Derivative of Dot Product of Vector-Valued Functions

Let: : $\mathbf r: x \mapsto \left\langle{r_1 \left({x}\right), r_2 \left({x}\right), \ldots, r_n \left({x}\right)}\right\rangle$ : $\mathbf q: x \mapsto \left\langle{q_1 \left({x}\right), q_2 \left({x}\right), \ldots, q_n \left({x}\right)}\right\rangle$ be differentiable vector-valued functions. The derivative of their dot product is given by: :$\dfrac \d {\d x} \left({\mathbf r \left({x}\right) \cdot \mathbf q \left({x}\right)}\right) = \mathbf r' \left({x}\right) \cdot \mathbf q \left({x}\right) + \mathbf r \left({x}\right) \cdot \mathbf q' \left({x}\right)$


Borel Sigma-Algebra of Subset is Trace Sigma-Algebra

Let $\left({X, \tau}\right)$ be a topological space, and let $A \subseteq X$ be a subset of $X$. Let $\tau_A$ be the subspace topology on $A$. Then the following equality of $\sigma$-algebras on $A$ holds: :$\mathcal B \left({A, \tau_A}\right) = \mathcal B \left({X, \tau}\right)_A$ where $\mathcal B$ signifies Borel $\sigma$-algebra, and $\mathcal B \left({X, \tau}\right)_A$ signifies trace $\sigma$-algebra.


Derivative of Vector Cross Product of Vector-Valued Functions

Let: :$\mathbf r: t \mapsto \begin {bmatrix} x \\ y \\ z \end{bmatrix}$ :$\mathbf q: t \mapsto \begin {bmatrix} \chi \\ \gamma \\ \zeta \end{bmatrix}$ be differentiable vector-valued functions, where: :$x, y, z, \chi, \gamma, \zeta$ are (images of) differentiable real functions. The derivative of the vector cross product of $\mathbf r$ and $\mathbf q$ is given by: :$D_t \left({\mathbf r \left({t}\right) \times \mathbf q \left({t}\right)}\right) = \mathbf r' \left({x}\right) \times \mathbf q \left({x}\right) + \mathbf r \left({x}\right) \times \mathbf q'\left({x}\right)$


Existence and Uniqueness of Monotone Class Generated by Collection of Subsets

Let $X$ be a set. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then $\mathfrak m \left({\mathcal G}\right)$, the monotone class generated by $\mathcal G$, exists and is unique.


Derivative of Product of Real Function and Vector-Valued Function

Let: :$\mathbf r:x \mapsto \mathbf z$ be a differentiable vector-valued function, where: :$\mathbf{z} = \begin{bmatrix} z_1 \\ z_2 \\ \vdots \\ z_n \end{bmatrix}$ such that: :$z_1, z_2, \cdots, z_n$ are (images of) differentiable real functions. Let: :$f: x \mapsto y$ be a differentiable real function. Then: :$D_x \left({y \, \mathbf z}\right) = \dfrac {\d y} {\d x} \mathbf z + y \dfrac {\d \mathbf z} {\d x}$


Generated Sigma-Algebra by Generated Monotone Class

Let $X$ be a set, and let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a nonempty collection of subsets of $X$. Suppose that $\mathcal G$ satisfies the following condition: :$(1):\quad A \in \mathcal G \implies \complement_X \left({A}\right) \in \mathcal G$ that is, $\mathcal G$ is closed under complement in $X$. Then: :$\mathfrak m \left({\mathcal G}\right) = \sigma \left({\mathcal G}\right)$ where $\mathfrak m$ denotes generated monotone class, and $\sigma$ denotes generated $\sigma$-algebra.


Heine-Borel Theorem/Metric Space

A metric space is compact {{iff}} it is both complete and totally bounded.


Borel Sigma-Algebra on Euclidean Space by Monotone Class

Let $\left({\R^n, \tau}\right)$ be the $n$-dimensional Euclidean space. Then: :$\mathcal B \left({\R^n, \tau}\right) = \mathfrak m \left({\tau}\right)$ where $\mathcal B$ denotes Borel $\sigma$-algebra, and $\mathfrak m$ denotes generated monotone class.


Measure is Strongly Additive

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Then $\mu$ is strongly additive, that is: : $\forall E, F \in \Sigma: \mu \left({E \cap F}\right) + \mu \left({E \cup F}\right) = \mu \left({E}\right) + \mu \left({F}\right)$


Measure is Subadditive

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Then $\mu$ is subadditive, that is: :$\forall E, F \in \Sigma: \mu \left({E \cup F}\right) \le \mu \left({E}\right) + \mu \left({F}\right)$


Representation of Degree One is Irreducible

Let $\left({G, \cdot}\right)$ be a finite group. Let $\rho: G \to \operatorname{GL} \left({V}\right)$ be a linear representation of $G$ on $V$ of degree $1$.  Then $\rho$ is an  irreducible linear representation.


Irreducible Representations of Abelian Group

Let $\left({G, \cdot}\right)$ be a finite abelian group. Let $V$ be a non-null vector space over an algebraically closed field $k$. Let $\rho: G \to \operatorname{GL} \left({V}\right)$ be a linear representation. Then $\rho$ is irreducible iff $\dim \left({V}\right) = 1$, where, $\dim$ denotes dimension.


Schur's Lemma (Representation Theory)

Let $\left({G, \cdot}\right)$ be a finite group. Let $V$ and $V'$ be two irreducible $G$-modules. Let $f: V \to V'$ be a homomorphism of $G$-modules. Then either: :$f \left({v}\right) = 0$ for all $v \in V$ or: : $f$ is an isomorphism.


Kernel is G-Module

Let $\struct {G, \cdot}$ be a group. Let $f: \struct {V, \phi} \to \struct {V', \mu}$ be a homomorphism of $G$-modules. Then its kernel $\map \ker f$ is a $G$-submodule of $V$.


Image is G-Module

Let $\left({G, \cdot}\right)$ be a group and let $f: \left({V, \phi}\right) \to \left({V', \mu}\right)$ be a homomorphism of $G$-modules. Then $\operatorname{Im} \left({f}\right)$ is a $G$-submodule of $V'$.


Set Intersection Preserves Subsets/Families of Sets

Let $I$ be an indexing set. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$. Let: :$\forall \beta \in I: A_\beta \subseteq B_\beta$ Then: :$\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in I} B_\alpha$


Set Intersection Preserves Subsets

Let $A, B, S, T$ be sets. Then: :$A \subseteq B, \ S \subseteq T \implies A \cap S \subseteq B \cap T$


Null Space Contains Zero Vector

Let: :$\map {\mathrm N} {\mathbf A} = \set {\mathbf x \in \R^n: \mathbf A \mathbf x = \mathbf 0}$ be the null space of $\mathbf A$, where: :$ \mathbf A_{m \times n} = \begin {bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \\ \end{bmatrix}$  is a matrix in the matrix space $\map {\MM_\R} {m, n}$. Then the null space of $\mathbf A$ contains the zero vector: :$\mathbf 0 \in \map {\mathrm N} {\mathbf A}$ where: :$\mathbf 0 = \mathbf 0_{m \times 1} = \begin {bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end {bmatrix}$


Commutative Linear Transformation is G-Module Homomorphism

Let $\rho: G \to \operatorname{GL} \left({V}\right)$ be a representation. Let $f: V \to V$ be a linear mapping. Let: :$\forall g \in G: \rho \left({g}\right) \circ f = f \circ \rho \left({g}\right)$ Then $f: V \to V$ is a $G$-module homomorphism. {{explain|between which $G$-modules? (probably $\left({G, \rho}\right) \to \left({G, \rho}\right)$, which makes it rather tautologous)}}


Characterization of Measures

Let $\struct {X, \Sigma}$ be a measurable space. Denote $\overline \R_{\ge 0}$ for the set of positive extended real numbers. A mapping $\mu: \Sigma \to \overline \R_{\ge 0}$ is a measure {{iff}}: :$(1):\quad \map \mu \O = 0$ :$(2):\quad \mu$ is finitely additive :$(3):\quad$ For every increasing sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\Sigma$, if $E_n \uparrow E$, then: ::::$\map \mu E = \ds \lim_{n \mathop \to \infty} \map \mu {E_n}$ where $E_n \uparrow E$ denotes limit of increasing sequence of sets. Alternatively, and equivalently, $(3)$ may be replaced by either of: :$(3'):\quad$ For every decreasing sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\Sigma$ for which $\map \mu {E_1}$ is finite, if $E_n \downarrow E$, then: ::::$\map \mu E = \ds \lim_{n \mathop \to \infty} \map \mu {E_n}$ :$(3''):\quad$ For every decreasing sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\Sigma$ for which $\map \mu {E_1}$ is finite, if $E_n \downarrow \O$, then: ::::$\ds \lim_{n \mathop \to \infty} \map \mu {E_n} = 0$ where $E_n \downarrow E$ denotes limit of decreasing sequence of sets.


Measure is Countably Subadditive

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Then $\mu$ is a countably subadditive function.


Dirac Measure is Measure

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $x \in X$, and let $\delta_x$ be the Dirac measure at $x$. Then $\delta_x$ is a measure.


Dirac Measure is Probability Measure

Let $\left({X, \mathcal A}\right)$ be a measurable space. Let $x \in X$, and let $\delta_x$ be the Dirac measure at $x$. Then $\delta_x$ is a probability measure.


Definition:Co-Countable Measure

Let $X$ be an uncountable set. Let $\Sigma$ be the $\sigma$-algebra of countable sets on $X$. Then the '''co-countable measure (on $X$)''' is the measure defined by: :$\mu: \Sigma \to \overline{\R}, \ \mu \left({E}\right) := \begin{cases} 0 & : \text{if $E$ is countable}\\ 1 & : \text{if $E$ is co-countable}\end{cases}$ where: : $\overline{\R}$ denotes the extended real numbers : $E$ is co-countable iff $X \setminus E$ is countable.


Co-Countable Measure is Measure

Let $X$ be an uncountable set. Let $\Sigma$ be the $\sigma$-algebra of countable sets on $X$. Then the co-countable measure $\mu$ on $X$ is a measure.


Counting Measure is Measure

Let $\left({X, \Sigma}\right)$ be a measurable space. Then the counting measure $\left\vert{\cdot}\right\vert$ on $\left({X, \Sigma}\right)$ is a measure.


Trivial Vector Space iff Zero Dimension

Let $V$ be a vector space. Then $V = \left\{{\mathbf 0}\right\}$ iff $\dim \left({V}\right) = 0$, where $\dim$ signifies dimension.


G-Submodule Test

Let $\left({V, \phi}\right)$ be a $G$-module over a field $k$. Let $W$ be a vector subspace of $V$. Then $\left({W, \phi_W}\right)$, where $\phi_W: G \times W \to W$ is the restriction of $\phi$ to $G \times W$, is a $G$-submodule of $V$ iff $\phi \left({G, W}\right) \subseteq W$.


Composite of Continuous Mappings is Continuous/Point

Let $T_1, T_2, T_3$ be topological spaces. Let the mapping $f : T_1 \to T_2$ be continuous at $x$. Let the mapping $g : T_2 \to T_3$ be continuous at $x$ at $\map f x$. Then the composite mapping $g \circ f : T_1 \to T_3$ is continuous at $x$ at $x$.


Null Space Closed under Vector Addition

Let: :$\map {\mathrm N} {\mathbf A} = \set {\mathbf x \in \R^n : \mathbf A \mathbf x = \mathbf 0}$ be the null space of $\mathbf A$, where: :$\mathbf A_{m \times n} = \begin {bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$, $\mathbf x_{n \times 1} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$ and $\mathbf 0_{m \times 1} = \begin {bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end {bmatrix}$  are matrices :the column matrix $\mathbf x_{n \times 1}$ is interpreted as a vector in $\R^n$. Then $\map {\mathrm N} {\mathbf A}$ is closed under vector addition: :$\forall \mathbf v, \mathbf w \in \map {\mathrm N} {\mathbf A}: \mathbf v + \mathbf w \in \map {\mathrm N} {\mathbf A}$


Null Space Closed under Scalar Multiplication

Let: :$\map {\mathrm N} {\mathbf A} = \set {\mathbf x \in \R^n : \mathbf {A x} = \mathbf 0}$ be the null space of $\mathbf A$, where: :$ \mathbf A_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end {bmatrix}$,  $\mathbf x_{n \times 1} = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end {bmatrix}$, $\mathbf 0_{m \times 1} = \begin {bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end {bmatrix}$ are matrices where each column is an element of a real vector space. Then $\map {\mathrm N} {\mathbf A}$ is closed under scalar multiplication: :$\forall \mathbf v \in \map {\mathrm N} {\mathbf A} ,\forall \lambda \in \R: \lambda \mathbf v \in \map {\mathrm N} {\mathbf A}$


Null Space is Subspace

Let: :$\operatorname{N} \left({\mathbf A}\right) = \left\{{\mathbf x \in \R^n: \mathbf {Ax} = \mathbf 0}\right\}$ be the null space of $\mathbf A$, where: :$ \mathbf A_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$,  $\mathbf x_{n \times 1} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf 0_{m \times 1} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$ are matrices. Then $\operatorname{N} \left({\mathbf A}\right)$ is a linear subspace of $\R^n$.


Norm is Continuous

Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space. Then the mapping $x \mapsto \norm x$ is continuous. Here, the metric used is the metric $d$ induced by $\norm {\,\cdot\,}$.


P-adic Valuation is Valuation

The $p$-adic valuation $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$ is a valuation on $\Q$.


P-adic Norm not Complete on Rational Numbers

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. Then: :the valued field $\struct {\Q, \norm {\,\cdot\,}_p}$ is not complete. That is, there exists a Cauchy sequence in $\struct {\Q, \norm{\,\cdot\,}_p}$ which does not converge to a limit in $\Q$.


P-adic Valuation on Integers is Valuation

Let $\nu_p^\Z: \Z \to \Z \cup \set {+\infty}$ be the $p$-adic valuation restricted to the integers. Then $\nu_p^\Z$ is a valuation.


Infinite Measure is Measure

Let $\struct {X, \Sigma}$ be a measurable space. Then the infinite measure $\mu$ on $\struct {X, \Sigma}$ is a measure.


Linear Combination of Measures

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\mu, \nu$ be measures on $\left({X, \Sigma}\right)$. Then for all positive real numbers $a, b \in \R_{\ge 0}$, the pointwise sum: :$a \mu + b \nu: \Sigma \to \overline \R, \ \left({a \mu + b \nu}\right) \left({E}\right) := a \mu \left({E}\right) + b \nu \left({E}\right)$ is also a measure on $\left({X, \Sigma}\right)$.


Series of Measures is Measure

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\left({\mu_n}\right)_{n \mathop \in \N}$ be a sequence of measures on $\left({X, \Sigma}\right)$. Let $\left({a_n}\right)_{n \mathop \in \N} \subseteq \R_{\ge 0}$ be a sequence of positive real numbers. Then the series of measures $\mu: \Sigma \to \overline{\R}$, defined by: :$\displaystyle \mu \left({E}\right) := \sum_{n \mathop \in \N} a_n \mu_n \left({E}\right)$ is also a measure on $\left({X, \Sigma}\right)$.


Quotient Set Determined by Relation Induced by Partition is That Partition

Let $S$ be a set. Let $\mathcal P$ be a partition of $S$. Let $\mathcal R$ be the relation induced by $\mathcal P$. Then the quotient set $S / \mathcal R$ of $S$ is $\mathcal P$ itself.


Null Sets Closed under Subset

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $N \in \Sigma$ be a $\mu$-null set, and let $M \in \Sigma$ be a subset of $N$. Then $M$ is also a $\mu$-null set.


Null Sets Closed under Countable Union

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\left({N_n}\right)_{n \in \N}$ be a sequence of $\mu$-null sets. Then $N := \displaystyle \bigcup_{n \mathop \in \N} N_n$ is also a $\mu$-null set.


Absolute Value induces Equivalence Compatible with Integer Multiplication

Let $\Z$ be the set of integers. Let $\RR$ be the relation on $\Z$ defined as: :$\forall x, y \in \Z: \struct {x, y} \in \RR \iff \size x = \size y$ where $\size x$ denotes the absolute value of $x$. Then $\RR$ is a congruence relation for integer multiplication.


Completion Theorem (Measure Spaces)

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Then there exists a completion $\left({X, \Sigma^*, \bar \mu}\right)$ of $\left({X, \Sigma, \mu}\right)$.


Restricted Measure is Measure

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\Sigma'$ be a sub-$\sigma$-algebra of $\Sigma$. Then the restricted measure $\mu \restriction_{\Sigma'}$ is a measure on the measurable space $\left({X, \Sigma'}\right)$.


Restricting Measure Preserves Finiteness

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\mu$ be a finite measure. Let $\Sigma'$ be a sub-$\sigma$-algebra of $\Sigma$. Then the restricted measure $\mu \restriction_{\Sigma'}$ is also a finite measure.


Central Subgroup is Normal

Let $G$ be a group. Let $H$ be a central subgroup of $G$. Then $H$ is a normal subgroup of $G$.


Angle Between Non-Zero Vectors Always Defined

The angle between two non-zero vectors in $\R^n$ is always defined.


Dynkin System Contains Empty Set

Let $X$ be a set, and let $\mathcal D$ be a Dynkin system on $X$. Then the empty set $\varnothing$ is an element of $\mathcal D$.


Dynkin System Closed under Disjoint Union

Let $X$ be a set, and let $\mathcal D$ be a Dynkin system on $X$. Let $D, E \in \mathcal D$ be disjoint. Then the union $D \cup E$ is also an element of $\mathcal D$.


Sigma-Algebra is Dynkin System

Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Then $\Sigma$ is a Dynkin system on $X$.


Existence and Uniqueness of Dynkin System Generated by Collection of Subsets

Let $X$ be a set. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then $\delta \left({\mathcal G}\right)$, the Dynkin system generated by $\mathcal G$, exists and is unique.


Generated Sigma-Algebra Contains Generated Dynkin System

Let $X$ be a set. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then $\delta \left({\mathcal G}\right) \subseteq \sigma \left({\mathcal G}\right)$. Here $\delta$ denotes generated Dynkin system, and $\sigma$ denotes generated $\sigma$-algebra.


Dynkin System Closed under Intersections is Sigma-Algebra

Let $X$ be a set, and let $\DD$ be a Dynkin system on $X$. Suppose that $\DD$ satisfies the following condition: :$(1):\quad \forall D, E \in \DD: D \cap E \in \DD$ That is, $\DD$ is closed under intersection. Then $\DD$ is a $\sigma$-algebra.


Dynkin System with Generator Closed under Intersection is Sigma-Algebra

Let $X$ be a set. Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$. Suppose that $\GG$ satisfies the following condition: :$(1):\quad \forall G, H \in \GG: G \cap H \in \GG$ That is, $\GG$ is closed under intersection. Then: :$\map \delta \GG = \map \sigma \GG$ where $\delta$ denotes generated Dynkin system, and $\sigma$ denotes generated $\sigma$-algebra.


Carathéodory's Theorem (Measure Theory)

Let $X$ be a set. Let $\SS \subseteq \powerset X$ be a semi-ring of subsets of $X$. Let $\mu: \SS \to \overline \R$ be a pre-measure on $\SS$. Let $\map \sigma \SS$ be the $\sigma$-algebra generated by $\SS$. Then $\mu$ extends to a measure $\mu^*$ on $\map \sigma \SS$. {{Refactor|extract corollary}}


Congruence Modulo Normal Subgroup is Congruence Relation

Let $\struct {G, \circ}$ be a group. Let $N$ be a normal subgroup of $G$. Then congruence modulo $N$ is a congruence relation for the group operation $\circ$.


Intersection Measure is Measure

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $F \in \Sigma$ be a measurable set. Then the intersection measure $\mu_F$ is a measure on the measurable space $\left({X, \Sigma}\right)$.


Uniqueness of Measures/Proof 1

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a generator for $\Sigma$; i.e., $\Sigma = \sigma \left({\mathcal G}\right)$. Suppose that $\mathcal G$ satisfies the following conditions: :$(1):\quad \forall G, H \in \mathcal G: G \cap H \in \mathcal G$ :$(2):\quad$ There exists an exhausting sequence $\left({G_n}\right)_{n \in \N} \uparrow X$ in $\mathcal G$ Let $\mu, \nu$ be measures on $\left({X, \Sigma}\right)$, and suppose that: :$(3):\quad \forall G \in \mathcal G: \mu \left({G}\right) = \nu \left({G}\right)$ :$(4):\quad \forall n \in \N: \mu \left({G_n}\right)$ is finite Then $\mu = \nu$. Alternatively, by Countable Cover induces Exhausting Sequence, the exhausting sequence in $(2)$ may be replaced by a countable $\mathcal G$-cover $\left({G_n}\right)_{n \in \N}$, still subject to $(4)$.


Lebesgue Measure Invariant under Translations

Let $\lambda^n$ be the $n$-dimensional Lebesgue measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$. Let $\mathbf x \in \R^n$. Then $\lambda^n$ is translation-invariant; i.e., for all $B \in \mathcal B \left({\R^n}\right)$, have: :$\lambda^n \left({\mathbf x + B}\right) = \lambda^n \left({B}\right)$ where $\mathbf x + B$ is the set $\left\{{\mathbf x + \mathbf b: \mathbf b \in B}\right\}$.


Translation-Invariant Measure on Euclidean Space is Multiple of Lebesgue Measure

Let $\mu$ be a measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$. Suppose that $\mu$ is translation-invariant. Also, suppose that $\kappa := \mu \left({\left[{0 \,.\,.\, 1}\right)^n }\right) < +\infty$. Then $\mu = \kappa \lambda^n$, where $\lambda^n$ is the $n$-dimensional Lebesgue measure.


Dynkin System Closed under Set Difference with Subset

Let $X$ be a set, and let $\mathcal D$ be a Dynkin system on $X$. Let $D, E \in \mathcal D$ and suppose that $E \subseteq D$. Then the set difference $D \setminus E$ is also an element of $\mathcal D$.


Coset Product of Normal Subgroup is Consistent with Subset Product Definition

Let $\struct {G, \circ}$ be a group. Let $N$ be a normal subgroup of $G$. Let $a, b \in G$. Let $a \circ N$ and $b \circ N$ be the left cosets of $a$ and $b$ by $N$. Then the coset product: :$\paren {a \circ N} \circ \paren {b \circ N} = \paren {a \circ b} \circ N$ is consistent with the definition of the coset product as the subset product of $a \circ N$ and $b \circ  N$: :$\paren {a \circ N} \paren {b \circ N} = \set {x \circ y: x \in a \circ N, y \in b \circ N}$


Congruence Relation iff Compatible with Operation

Let $\struct {S, \circ}$ be an algebraic structure. Let $\RR$ be an equivalence relation on $S$. Then $\RR$ is a congruence relation for $\circ$ {{iff}}: {{begin-eqn}} {{eqn | ll= \forall x, y, z \in S:       | l = x \mathrel \RR y       | o = \implies       | r = \paren {x \circ z} \mathrel \RR \paren {y \circ z} }} {{eqn | l = x \mathrel \RR y       | o = \implies       | r = \paren {z \circ x} \mathrel \RR \paren {z \circ y} }} {{end-eqn}} That is, {{iff}} $\RR$ is compatible with $\circ$.


Outer Measure of Limit of Increasing Sequence of Sets

Let $\mu^*$ be an outer measure on a set $X$. Let $\left\langle{S_n}\right\rangle$ be an increasing sequence of $\mu^*$-measurable sets, and let $S_n \uparrow S$ (as $n \to \infty$). Then for any subset $A \subseteq X$: : $\displaystyle \mu^* \left({A \cap S}\right) = \lim_{n \to \infty} \mu^* \left({A \cap S_n}\right)$


Half-Open Rectangles form Semiring of Sets

The half-open $n$-rectangles $\JJ_{ho}^n$ form a semiring of sets.


Lebesgue Pre-Measure is Pre-Measure

The Lebesgue pre-measure $\lambda^n$ on the half-open $n$-rectangles $\mathcal{J}_{ho}^n$ is a pre-measure.


Existence and Uniqueness of Lebesgue Measure

Let $\lambda^n$ be the Lebesgue pre-measure on the half-open $n$-rectangles $\mathcal{J}_{ho}^n$. Then Lebesgue measure, the extension of $\lambda^n$ to the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$, exists and is unique.


Lebesgue Measure is Diffuse

Let $\lambda^n$ be Lebesgue measure on $\R^n$. Then $\lambda^n$ is a diffuse measure.


Half-Open Rectangles Closed under Intersection

Let $\lefthalf-open $n$-rectangles. Then $\lefthalf-open $n$-rectangle.


Equivalence of Definitions of Null Set in Euclidean Space

Let $\lambda^n$ be $n$-dimensional Lebesgue measure on $\R^n$. Let $E \subseteq \R^n$. Then the following are equivalent: :$(1):\quad \exists B \in \mathcal B \left({\R^n}\right): E \subseteq B, \lambda^n \left({B}\right) = 0$ :$(2):\quad$ For every $\epsilon > 0$, there exists a countable cover $\left({J_i}\right)_{i \mathop \in \N}$ of $E$ by open $n$-rectangles, such that: ::$\displaystyle \sum_{i \mathop = 1}^\infty \operatorname{vol} \left({J_i}\right) \le \epsilon$ {{explain|link to some definition (might not exist) for $\operatorname{vol}$}}


Homomorphism to Group Preserves Inverses

Let $\struct {S, \circ}$ be an algebraic structure. Let $\struct {T, *}$ be a group. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a homomorphism. Let $\struct {S, \circ}$ have an identity $e_S$. Let $x^{-1}$ be an inverse of $x$ for $\circ$. Then $\map \phi {x^{-1} }$ is an inverse of $\map \phi x$ for $*$.


Equivalence Induced by Epimorphism is Congruence Relation

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures. Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an epimorphism. Let $\mathcal R_\phi$ be the equivalence induced by $\phi$. Then the induced equivalence $\mathcal R_\phi$ is a congruence relation for $\circ$.


Unique Isomorphism from Quotient Mapping to Epimorphism Domain

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an epimorphism. Let $\mathcal R_\phi$ be the equivalence induced by $\phi$. Let $S / \mathcal R_\phi$ be the quotient of $S$ by $\mathcal R_\phi$. Let $q_{\mathcal R_\phi}: S \to S / \mathcal R_\phi$ be the quotient mapping induced by $\mathcal R_\phi$. Let $\struct {S / \mathcal R_\phi}, {\circ_{\mathcal R_\phi} }$ be the quotient structure defined by $\mathcal R_\phi$. Then there is one and only one isomorphism: :$\psi: \struct {S / \mathcal R_\phi}, {\circ_{\mathcal R_\phi} } \to \struct {T, *}$ which satisfies: :$\psi \bullet q_{\mathcal R_\phi} = \phi$ where, in order not to cause notational confusion, $\bullet$ is used as the symbol to denote composition of mappings.


Identity is in Kernel of Group Homomorphism

Let $G$ and $H$ be groups. Let $e_G$ and $e_H$ be the identity elements of $G$ and $H$ respectively. Let $\phi: G \to H$ be a (group) homomorphism from $G$ to $H$. Then: :$e_G \in \map \ker \phi$ where $\map \ker \phi$ is the kernel of $\phi$.


Union is Dominated by Disjoint Union

Let $I$ be an indexing set. For all $i \in I$, let $S_i$ be a set. Then: :$\displaystyle \bigcup_{i \mathop \in I} S_i \preccurlyeq \bigsqcup_{i \mathop \in I} S_i$ where $\preccurlyeq$ denotes domination, $\bigcup$ denotes union, and $\bigsqcup$ denotes disjoint union.


Disjoint Union Preserves Domination

Let $I$ be an indexing set. For all $i \in I$, let $A_i$ and $B_i$ be sets such that $A_i \preccurlyeq B_i$. Here, $\preccurlyeq$ denotes domination. Then: :$\displaystyle \bigsqcup_{i \mathop \in I} A_i \preccurlyeq \bigsqcup_{i \mathop \in I} B_i$ where $\bigsqcup$ denotes disjoint union.


Epimorphism from Real Numbers to Circle Group

Let $\struct {K, \times}$ be the circle group, that is: :$K = \set {z \in \C: \cmod z = 1}$ under complex multiplication. Let $f: \R \to K$ be the mapping from the real numbers to $K$ defined as: :$\forall x \in \R: \map f x = \cos x + i \sin x$ Then $f: \struct {\R, +} \to \struct {K, \times}$ is a group epimorphism. Its kernel is: :$\map \ker f = \set {2 \pi n: n \in \Z}$


Increasing Sequence of Sets induces Partition on Limit

Let $\sequence {S_n}_{n \mathop \in \N} \uparrow S$ be an increasing sequence of sets with limit $S$. Define $T_1 = S_1$, and, for $n \in \N$, $T_{n + 1} = S_{n + 1} \setminus S_n$, where $\setminus$ denotes set difference. Then $\sequence {T_n}_{n \mathop \in \N}$ is a countable partition of $S$.


Power Function on Complex Numbers is Epimorphism

Let $n \in \Z_{>0}$ be a strictly positive integer. Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers. Let $f_n: \C_{\ne 0} \to \C_{\ne 0}$ be the mapping from the set of complex numbers less zero to itself defined as: :$\forall z \in \C_{\ne 0}: \map {f_n} z = z^n$ Then $f_n: \struct {\C_{\ne 0}, \times} \to \struct {\C_{\ne 0}, \times}$ is a group epimorphism. The kernel of $f_n$ is the set of complex $n$th roots of unity.


Real Part as Mapping is Endomorphism for Complex Addition

Let $\struct {\C, +}$ be the additive group of complex numbers. Let $\struct {\R, +}$ be the additive group of real numbers. Let $f: \C \to \R$ be the mapping from the complex numbers to the real numbers defined as: :$\forall z \in \C: \map f z = \map \Re z$ where $\map \Re z$ denotes the real part of $z$. Then $f: \struct {\C, +} \to \struct {\R, +}$ is a group epimorphism. Its kernel is the set: :$\map \ker f = \set {i x: x \in \R}$ of wholly imaginary numbers.


Equivalence of Definitions of Semiring of Sets

{{TFAE|def = Semiring of Sets}} A collection $\SS$ of subsets of a set $X$ is a semiring (of sets) {{iff}}: :$(1):\quad \O \in \SS$ :$(2):\quad A, B \in \SS \implies A \cap B \in \SS$; that is, $\SS$ is $\cap$-stable :$(3):\quad$ If $A, A_1 \in \SS$ such that $A_1 \subseteq A$, then there exists a finite sequence $A_2, A_3, \ldots, A_n \in \SS$ such that: ::$(3a):\quad \displaystyle A = \bigcup_{k \mathop = 1}^n A_k$ ::$(3b):\quad$ The $A_k$ are pairwise disjoint We prove that criterion $(3)$ can be replaced by: :$(3'):\quad$ If $A, B \in \SS$, then there exist finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \SS$ such that $\displaystyle A \setminus B = \bigcup_{k \mathop = 1}^n A_k$.


Imaginary Part as Mapping is Endomorphism for Complex Addition

Let $\struct {\C, +}$ be the additive group of complex numbers. Let $\struct {\R, +}$ be the additive group of real numbers. Let $f: \C \to \R$ be the mapping from the complex numbers to the real numbers defined as: :$\forall z \in \C: \map f z = \map \Im z$ where $\map \Im z$ denotes the imaginary part of $z$. Then $f: \struct {\C, +} \to \struct {\R, +}$ is a group epimorphism. Its kernel is the set: :$\map \ker f = \R$ of (wholly) real numbers.


Reduced Echelon Matrix is Unique

Every $m \times n$ matrix is row equivalent to exactly one $m \times n$ reduced echelon matrix. That is, the reduced echelon form of a matrix is unique.


Measure of Set Difference with Subset

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $S, T \in \Sigma$ be such that $S \subseteq T$, and suppose that $\mu \paren S < +\infty$. Then: :$\mu \paren {T \setminus S} = \mu \paren T - \mu \paren S$ where $T \setminus S$ denotes set difference.


Subset of Codomain is Superset of Image of Preimage

Let $f: S \to T$ be a mapping. Then: :$B \subseteq T \implies \paren {f \circ f^{-1} } \sqbrk B \subseteq B$ where: :$f \sqbrk B$ denotes the image of $B$ under $f$ :$f^{-1}$ denotes the inverse of $f$ :$f \circ f^{-1}$ denotes composition of $f$ and $f^{-1}$. This can be expressed in the language and notation of direct image mappings and inverse image mappings as: :$\forall B \in \powerset T: \map {\paren {f^\to \circ f^\gets} } B \subseteq B$


Induced Outer Measure Restricted to Semiring is Pre-Measure

Let $\mathcal S$ be a semiring over a set $X$. Let $\mu: \mathcal S \to \overline \R_{\ge 0}$ be a pre-measure on $\mathcal S$, where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers. Let $\mu^*: \mathcal P \left({X}\right) \to \overline \R_{\ge 0}$ be the outer measure induced by $\mu$. Then: :$\displaystyle \mu^*\restriction_{\mathcal S} \, = \mu$ where $\restriction$ denotes restriction.


Ordering on Extended Real Numbers is Ordering

Denote with $\le$ the usual ordering on the extended real numbers $\overline \R$. Then $\le$ is an ordering, and so $\overline \R$ is an ordered set.


Ordering on Extended Real Numbers is Total Ordering

Let $\le$ denote the ordering on the extended real numbers $\overline \R$. Then $\le$ is a total ordering, and so $\overline \R$ is a toset.


Extended Real Number Space is Compact

The extended real number space is compact.


Euclidean Space is Subspace of Extended Real Number Space

Let $\struct {\overline \R, \tau}$ be the extended real number space. Then $\tau {\restriction_\R}$, the subspace topology on $\R$, is the Euclidean topology. That is, Euclidean $1$-space is a subspace of the extended real number space.


Lagrange's Formula

Let: :$\mathbf a = \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix}$, $\mathbf b = \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix}$, $\mathbf c = \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix}$ be vectors in a vector space of $3$ dimensions. Then: :$\mathbf a \times \paren {\mathbf b \times \mathbf c} = \paren {\mathbf a \cdot \mathbf c} \mathbf b - \paren {\mathbf a \cdot \mathbf b} \mathbf c$


External Direct Product of Groups is Group/Finite Product

The external direct product of a finite sequence of groups is itself a group.


External Direct Product Inverses/General Result

Let $\displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$. Let $\left({x_1, x_2, \ldots, x_n}\right) \in S$. Let $y_k$ be an inverse of $x_k$ in $\left({S_k, \circ_k}\right)$ for each of $k \in \N^*_n$. Then $\left({y_1, y_2, \ldots, y_n}\right)$ is the inverse of $\left({x_1, x_2, \ldots, x_n}\right) \in S$ in $\left({S, \circ}\right)$.


External Direct Product Identity/General Result

Let $\displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$. Let $e_1, e_2, \ldots, e_n$ be the identity elements of $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$ respectively. Then $\left({e_1, e_2, \ldots, e_n}\right)$ is the identity element of $\left({S, \circ}\right)$.


External Direct Product Commutativity/General Result

Let $\displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$. If $\circ_1, \ldots, \circ_n$ are all commutative, then so is $\circ$.


External Direct Product Associativity/General Result

Let $\displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$. If $\circ_1, \ldots, \circ_n$ are all associative, then so is $\circ$.


Vector Cross Product is not Associative

The vector cross product is ''not'' associative. That is, in general: :$\mathbf a \times \left({\mathbf b \times \mathbf c}\right) \ne \left({\mathbf a \times \mathbf b}\right) \times \mathbf c$ for $\mathbf {a}, \mathbf {b}, \mathbf {c} \in \R^3$.


Vector Cross Product is Anticommutative

The vector cross product is anticommutative: :$\forall \mathbf a, \mathbf b \in \R^3: \mathbf a \times \mathbf b = -\left({\mathbf b \times \mathbf a}\right)$


Canonical Injection is Monomorphism/General Result

Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \dotsc, \struct {S_j, \circ_j}, \dotsc, \struct {S_n, \circ_n}$ be algebraic structures with identities $e_1, e_2, \dotsc, e_j, \dotsc, e_n$ respectively. Then the canonical injection: :$\displaystyle \inj_j: \struct {S_j, \circ_j} \to \prod_{i \mathop = 1}^n \struct {S_i, \circ_i}$ defined as: :$\map {\inj_j} x = \tuple {e_1, e_2, \dotsc, e_{j - 1}, x, e_{j + 1}, \dotsc, e_n}$ is a monomorphism.


Orthogonality of Solutions to the Sturm-Liouville Equation with Distinct Eigenvalues

Let $f \left({x}\right)$ and $g \left({x}\right)$ be solutions of the Sturm-Liouville equation: :$(1): \quad -\dfrac {\mathrm d} {\mathrm d x} \left({p \left({x}\right) \dfrac {\mathrm d y} {\mathrm d x}}\right) + q \left({x}\right) y = \lambda w \left({x}\right) y$ where $y$ is a function of the free variable $x$. The functions $p \left({x}\right)$, $q \left({x}\right)$ and $w \left({x}\right)$ are specified. In the simplest cases they are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$. In addition: :$(1a): \quad p \left({x}\right) > 0$ has a continuous derivative :$(1b): \quad w \left({x}\right) > 0$ :$(1c): \quad y$ is typically required to satisfy some boundary conditions at $a$ and $b$. Assume that the Sturm-Liouville problem is regular, that is, $p \left({x}\right)^{-1} > 0$, $q \left({x}\right)$, and $w \left({x}\right) > 0$ are real-valued integrable functions over the closed interval  $\left[{a \,.\,.\, b}\right]$, with ''separated boundary conditions'' of the form:  :$(2): \quad y \left({a}\right) \cos \alpha - p \left({a}\right) y' \left({a}\right)\sin \alpha = 0$ :$(3): \quad y \left({b}\right) \cos \beta - p \left({b}\right) y' \left({b}\right)\sin \beta = 0$ where $\alpha, \beta \in \left[{0 \,.\,.\, \pi}\right)$. Then: :$\displaystyle \left\langle{f, g}\right\rangle = \int_a^b \overline {f \left({x}\right)} q \left({x}\right) w \left({x}\right) \, \mathrm d x = 0$ where $f \left({x}\right)$ and $g \left({x}\right)$ are solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues and $w \left({x}\right)$ is the "weight" or "density" function.


Row Equivalent Matrix for Homogeneous System has same Solutions

Let $\mathbf A$ be a matrix in the matrix space $\map {\MM_\R} {m, n}$ such that: :$\mathbf A \mathbf x = \mathbf 0$ represents a homogeneous system of linear equations. Let $\mathbf H$ be row equivalent to $\mathbf A$. Then the solution set of $\mathbf H \mathbf x = \mathbf 0$ equals the solution set of $\mathbf A \mathbf x = \mathbf 0$. That is: :$\mathbf A \sim \mathbf H \implies \set {\mathbf x: \mathbf A \mathbf x = \mathbf 0} = \set {\mathbf x: \mathbf H \mathbf x = \mathbf 0}$ where $\sim$ represents row equivalence.


Null Space of Reduced Echelon Form

Let $\mathbf A$ be a matrix in the matrix space $\map {\MM_\R} {m, n}$ such that: :$\mathbf A \mathbf x = \mathbf 0$ represents a homogeneous system of linear equations. The null space of $\mathbf A$ is the same as that of the null space of the reduced row echelon form of $\mathbf A$: :$\map {\mathrm N} {\mathbf A} = \map {\mathrm N} {\map {\mathrm {rref} } {\mathbf A} }$


Null Space Contains Only Zero Vector iff Columns are Independent

Let: {{begin-eqn}} {{eqn | l = \mathbf A_{m \times n}       | r = \begin{bmatrix} \mathbf a_1 & \mathbf a_2 & \cdots & \mathbf a_n \end{bmatrix} }} {{end-eqn}} be a matrix where: :$\forall i: 1 \le i \le n: \mathbf a_i = \begin{bmatrix} a_{1i} \\ a_{2i} \\ \vdots \\ a_{mi} \end{bmatrix} \in \R^m$ are vectors. Then: :$\set {\mathbf a_1, \mathbf a_2, \cdots, \mathbf a_n}$ is a linearly independent set  {{iff}}: :$\map {\mathrm N} {\mathbf A} = \set {\mathbf 0_{n \times 1} }$ where $\map {\mathrm N} {\mathbf A}$ is the null space of $\mathbf A$.


Axiom of Dependent Choice Implies Axiom of Countable Choice

The axiom of dependent choice implies the axiom of countable choice.


Axiom of Choice Implies Axiom of Dependent Choice

The axiom of choice implies the axiom of dependent choice.


Infimum of Empty Set is Greatest Element

Let $\struct {S, \preceq}$ be an ordered set. Suppose that $\map \inf \O$, the infimum of the empty set, exists in $S$. Then $\forall s \in S: s \preceq \map \inf \O$. That is, $\map \inf \O$ is the greatest element of $S$.


Supremum of Empty Set is Smallest Element

Let $\struct {S, \preceq}$ be an ordered set. Then: :the supremum of the empty set exists {{iff}} the smallest element of $S$ exists in which case: :$\map \sup \O$ is the smallest element of $S$


Countable Union of Countable Sets is Countable

Let the axiom of countable choice be accepted. Then it can be proved that a countable union of countable sets is countable.


Extended Real Addition is Commutative

Extended real addition $+_{\overline{\R}}$ is commutative. That is, for all $x, y \in \overline{\R}$: :$x +_{\overline{\R}} y = y +_{\overline{\R}} x$ whenever at least one of the sides is defined.


Extended Real Addition is Associative

Extended real addition $+_{\overline{\R}}$ is commutative. That is, for all $x, y, z \in \overline{\R}$: :$(1):\quad x +_{\overline{\R}} \left({y +_{\overline{\R}} z}\right) = \left({x +_{\overline{\R}} y}\right) +_{\overline{\R}} z$ whenever at least one of the sides is defined.


Extended Real Multiplication is Commutative

Extended real multiplication $\times_{\overline \R}$ is commutative. That is, for all $x, y \in \overline \R$: :$x \times_{\overline \R} y = y \times_{\overline \R} x$


Extended Real Multiplication is Associative

Extended real multiplication $\cdot_{\overline \R}$ is commutative. That is, for all $x, y, z \in \overline \R$: :$(1): \quad x \cdot_{\overline \R} \left({y \cdot_{\overline \R} z}\right) = \left({x \cdot_{\overline \R} y}\right) \cdot_{\overline \R} z$


Extended Real Numbers under Multiplication form Monoid

Denote with $\overline \R$ the extended real numbers. Denote with $\cdot_{\overline \R}$ the extended real multiplication. The algebraic structure $\struct {\overline \R, \cdot_{\overline \R} }$ is a monoid.


Extended Real Numbers under Multiplication form Commutative Monoid

Denote with $\overline \R$ the extended real numbers. Denote with $\cdot_{\overline \R}$ the extended real multiplication. The algebraic structure $\left({\overline \R, \cdot_{\overline \R}}\right)$ is a commutative monoid.


Translation in Euclidean Space is Measurable Mapping

Let $\mathcal B$ be the Borel $\sigma$-algebra on $\R^n$. Let $\mathbf x \in \R^n$, and denote with $\tau_{\mathbf x}: \R^n \to \R^n$ translation by $\mathbf x$. Then $\tau_{\mathbf x}$ is $\mathcal B \, / \, \mathcal B$-measurable.


Internal Group Direct Product Commutativity/General Result

Let $\struct {G, \circ}$ be the internal group direct product of $H_1, H_2, \ldots, H_n$. Let $h_i$ and $h_j$ be elements of $H_i$ and $H_j$ respectively, $i \ne j$. Then $h_i \circ h_j = h_j \circ h_i$.


Mapping Measurable iff Measurable on Generator

Let $\left({X, \Sigma}\right)$ and $\left({X', \Sigma'}\right)$ be measurable spaces. Suppose that $\Sigma'$ is generated by $\mathcal{G}'$. Then a mapping $f: X \to X'$ is $\Sigma \, / \, \Sigma'$-measurable iff: :$\forall G' \in \mathcal{G}': f^{-1} \left({G'}\right) \in \Sigma$ That is, iff the preimage of every generator under $f$ is a measurable set.


Internal Direct Product Theorem/General Result

Let $G$ be a group whose identity is $e$. Let $\sequence {H_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of subgroups of $G$. Then $G$ is the internal group direct product of $\sequence {H_k}_{1 \mathop \le k \mathop \le n}$ {{iff}}: :$(1): \quad G = H_1 H_2 \cdots H_n$ :$(2): \quad \sequence {H_k}_{1 \mathop \le k \mathop \le n}$ is a sequence of independent subgroups :$(3): \quad \forall k \in \set {1, 2, \ldots, n}: H_k \lhd G$ where $H_k \lhd G$ denotes that $H_k$ is a normal subgroup of $G$.


Structure Induced by Group Operation is Group

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $S$ be a set. Let $\struct {G^S, \oplus}$ be the structure on $G^S$ induced by $\circ$. Then $\struct {G^S, \oplus}$ is a group.


Continuous Mapping is Measurable

Let $\left({X, \tau}\right)$ and $\left({X', \tau'}\right)$ be topological spaces. Denote with $\mathcal B \left({X, \tau}\right)$ and $\mathcal B \left({X', \tau'}\right)$ their respective Borel $\sigma$-algebras. Let $f: X \to X'$ be a continuous mapping. Then $f$ is $\mathcal B \left({X, \tau}\right) \, / \, \mathcal B \left({X', \tau'}\right)$-measurable.


Composition of Measurable Mappings is Measurable

Let $\struct {X_1, \Sigma_1}$, $\struct {X_2, \Sigma_2}$ and $\struct {X_3, \Sigma_3}$ be measurable spaces. Let $f: X_1 \to X_2$ be a $\Sigma_1 \, / \, \Sigma_2$-measurable mapping. Let $g: X_2 \to X_3$ be a $\Sigma_2 \, / \, \Sigma_3$-measurable mapping. Then their composition $g \circ f: X_1 \to X_3$ is $\Sigma_1 \, / \, \Sigma_3$-measurable.


Restriction of Equivalence Relation is Equivalence

Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be an equivalence relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\mathcal R {\restriction_T} \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$. Then $\mathcal R {\restriction_T}$ is an equivalence relation on $T$.


Restriction of Reflexive Relation is Reflexive

Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be a reflexive relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\mathcal R {\restriction_T} \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$. Then $\mathcal R {\restriction_T}$ is a reflexive relation on $T$.


Restriction of Symmetric Relation is Symmetric

Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be a symmetric relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\mathcal R {\restriction_T} \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$. Then $\mathcal R {\restriction_T}$ is a symmetric relation on $T$.


Restriction of Transitive Relation is Transitive

Let $S$ be a set. Let $\RR \subseteq S \times S$ be a transitive relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$. Then $\RR {\restriction_T}$ is a transitive relation on $T$.


Restriction of Asymmetric Relation is Asymmetric

Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be a asymmetric relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\mathcal R \restriction_T \ \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$. Then $\mathcal R \restriction_T$ is a asymmetric relation on $T$.


Restriction of Antisymmetric Relation is Antisymmetric

Let $S$ be a set. Let $\RR \subseteq S \times S$ be an antisymmetric relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$. Then $\RR {\restriction_T}$ is an antisymmetric relation on $T$.


Restriction of Antireflexive Relation is Antireflexive

Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be an antireflexive relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\mathcal R \restriction_T \ \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$. Then $\mathcal R \restriction_T$ is an antireflexive relation on $T$.


Restriction of Antitransitive Relation is Antitransitive

Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be an antitransitive relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\mathcal R \restriction_T \ \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$. Then $\mathcal R \restriction_T$ is an antitransitive relation on $T$.


Relation is Total iff Union with Inverse is Trivial Relation

Let $\mathcal R$ be a relation on $S$. Then $\mathcal R$ is a total relation {{iff}}: :$\mathcal R \cup \mathcal R^{-1} = S \times S$ where: : $\mathcal R^{-1}$ is the inverse of $\mathcal R$. : $S \times S$ is the trivial relation on $S$.


Restriction of Connected Relation is Connected

Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be a connected relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\mathcal R \restriction_T \ \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$. Then $\mathcal R \restriction_T$ is a connected relation on $T$.


Restriction of Ordering is Ordering

Let $S$ be a set. Let $\preceq$ be an ordering on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\preceq \restriction_T$ be the restriction of $\preceq$ to $T$. Then $\preceq \restriction_T$ is an ordering on $T$.


Restriction of Serial Relation is Not Necessarily Serial

Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be a serial relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\mathcal R \restriction_T \ \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$. Then $\mathcal R \restriction_T$ is not necessarily a serial relation on $T$.


Restriction of Non-Reflexive Relation is Not Necessarily Non-Reflexive

Let $S$ be a set. Let $\RR \subseteq S \times S$ be a non-reflexive relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$. Then $\RR {\restriction_T}$ is not necessarily a non-reflexive relation on $T$.


Restriction of Non-Symmetric Relation is Not Necessarily Non-Symmetric

Let $S$ be a set. Let $\RR \subseteq S \times S$ be a non-symmetric relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\RR {\restriction_T} \ \subseteq T \times T$ be the restriction of $\RR$ to $T$. Then $\RR {\restriction_T}$ is not necessarily a non-symmetric relation on $T$.


Restriction of Non-Transitive Relation is Not Necessarily Non-Transitive

Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be a non-transitive relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\mathcal R \restriction_T \ \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$. Then $\mathcal R \restriction_T$ is not necessarily a non-transitive relation on $T$.


Existence and Uniqueness of Sigma-Algebra Generated by Collection of Mappings

Let $\left({X_i, \Sigma_i}\right)$ be measurable spaces, with $i \in I$ for some index set $I$. Let $X$ be a set, and let, for $i \in I$, $f_i: X \to X_i$ be a mapping. Then $\sigma \left({f_i: i \in I}\right)$, the $\sigma$-algebra generated by $\left({f_i}\right)_{i \in I}$, exists and is unique.


Characterization of Sigma-Algebra Generated by Collection of Mappings

Let $\left({X_i, \Sigma_i}\right)$ be measurable spaces, with $i \in I$ for some index set $I$. Let $X$ be a set, and let, for $i \in I$, $f_i: X \to X_i$ be a mapping. Then: :$\sigma \left({f_i: i \in I}\right) = \sigma \left({\displaystyle \bigcup_{i \mathop \in I} f_i^{-1} \left({\Sigma_i}\right)}\right)$ where: :$\sigma \left({f_i: i \in I}\right)$ is the $\sigma$-algebra generated by $\left({f_i}\right)_{i \in I}$ :$\sigma \left({\displaystyle \bigcup_{i \mathop \in I} f_i^{-1} \left({\Sigma_i}\right)}\right)$ is the $\sigma$-algebra generated by $\displaystyle \bigcup_{i \mathop \in I} f_i^{-1} \left({\Sigma_i}\right)$ :$f_i^{-1} \left({\Sigma_i}\right)$ denotes the pre-image $\sigma$-algebra on $X$ by $f$


Restriction of Non-Connected Relation is Not Necessarily Non-Connected

Let $S$ be a set. Let $\RR \subseteq S \times S$ be a non-connected relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$. Then $\RR {\restriction_T}$ is not necessarily a non-connected relation on $T$.


Duals of Isomorphic Ordered Sets are Isomorphic

Let $\struct {S, \preccurlyeq_1}$ and $\struct {T, \preccurlyeq_2}$ be ordered sets. Let $\struct {S, \succcurlyeq_1}$ and $\struct {T, \succcurlyeq_2}$ be the dual ordered sets of $\struct {S, \preccurlyeq_1}$ and $\struct {T, \preccurlyeq_2}$ respectively. Let $f: \struct {S, \preccurlyeq_1} \to \struct {T, \preccurlyeq_2}$ be an order isomorphism. Then $f: \struct {S, \succcurlyeq_1} \to {T, \succcurlyeq_2} $ is also an order isomorphism.


Pushforward Measure is Measure

Let $\struct {X, \Sigma}$ and $\struct {X', \Sigma'}$ be measurable spaces. Let $\mu$ be a measure on $\struct {X, \Sigma}$. Let $f: X \to X'$ be a $\Sigma \, / \, \Sigma'$-measurable mapping. Then the pushforward measure $f_* \mu: \Sigma' \to \overline \R$ is a measure.


Lebesgue Measure Invariant under Orthogonal Group

Let $M \in \map {\mathrm O} {n, \R}$ be an orthogonal matrix. Let $\lambda^n$ be $n$-dimensional Lebesgue measure. Then the pushforward measure $M_* \lambda^n$ equals $\lambda^n$.


Pushforward of Lebesgue Measure under General Linear Group

Let $M \in \GL {n, \R}$ be an invertible matrix. Let $\lambda^n$ be $n$-dimensional Lebesgue measure. Then the pushforward measure $M_* \lambda^n$ satisfies: :$M_* \lambda^n = \size {\det M^{-1} } \cdot \lambda^n$


Pre-Image Sigma-Algebra on Domain is Generated by Mapping

Let $X, X'$ be sets, and let $f: X \to X'$ be a mapping. Let $\Sigma'$ be a $\sigma$-algebra on $X'$. Let $f: X \to X'$ be a mapping. Then: :$\sigma \left({f}\right) = f^{-1} \left({\Sigma'}\right)$ where :$\sigma \left({f}\right)$ denotes the $\sigma$-algebra generated by $f$ :$f^{-1} \left({\Sigma'}\right)$ denotes the pre-image $\sigma$-algebra under $f$


Mapping between Euclidean Spaces Measurable iff Components Measurable

Let $\R^n$ and $\R^m$ be Euclidean spaces. Denote by $\mathcal{B}^n$ and $\mathcal{B}^m$ their respective Borel $\sigma$-algebras. Denote with $\mathcal B$ the Borel $\sigma$-algebra on $\R$. Let $f: \R^n \to \R^m$ be a mapping, and write: :$f \left({\mathbf x}\right) = \begin{bmatrix}f_1 \left({\mathbf x}\right) \\ \vdots \\ f_m \left({\mathbf x}\right)\end{bmatrix}$ with, for $1 \le i \le m$, $f_i: \R^n \to \R$. Then $f$ is $\mathcal{B}^n \, / \, \mathcal{B}^m$-measurable iff: :$\forall i:f_i: \R^n \to \R$ is $\mathcal{B}^n \, / \, \mathcal B$-measurable


Pre-Image Sigma-Algebra of Generated Sigma-Algebra

Let $f: X \to Y$ be a mapping. Let $\mathcal G \subseteq \mathcal P \left({Y}\right)$ be a collection of subsets of $Y$. Then the following equality of $\sigma$-algebras on $X$ holds: :$f^{-1} \left({\sigma \left({\mathcal G}\right)}\right) = \sigma \left({f^{-1} \left({\mathcal G}\right)}\right)$ where :$\sigma$ denotes a generated $\sigma$-algebra :$f^{-1} \left({\sigma \left({\mathcal G}\right)}\right)$ denotes the pre-image $\sigma$-algebra :$f^{-1} \left({\mathcal G}\right)$ is the preimage of $\mathcal G$ under $f$


Stieltjes Function of Measure is Stieltjes Function

Let $\mu$ be a measure on $\R$ with the Borel $\sigma$-algebra $\mathcal B \left({\R}\right)$. Suppose that for every $n \in \N$: :$\mu \left[{-n \,.\,.\, n}\right] < +\infty$ Then $F_\mu: \R \to \overline \R$, the Stieltjes function of $\mu$, is a Stieltjes function.


Characterization of Differentiability

{{refactor|one-dimensional case deserves its own page, perhaps}} Let $\mathbb X$ be an open rectangle of $\R^n$.  Let $f: \mathbb X \to \R, \mathbf x \mapsto \map f {\mathbf x}$ be a real-valued function. Let $\mathbf x = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end {bmatrix} \in \R^n$. Let $\map {\Delta f} {\mathbf x} = \map f {\mathbf x + \Delta \mathbf x} - \map f {\mathbf x}$ . Let $\dfrac {\partial f} {\partial x_j}$ be the partial derivative of $f$ {{WRT|Differentiation}} $x_j$. Then $f$ is differentiable {{iff}} there exists some $\map {\Delta f} {\mathbf x}$ such that: :$\map {\Delta  f} {\mathbf x} = \displaystyle \sum_{i \mathop = 1}^n \frac {\partial \map f {\mathbf x} } {\partial x_i} \Delta x_i + \sum_{i \mathop = 1}^n \varepsilon_i \Delta x_i$ where $\forall i: 1 \le i \le n: \varepsilon_i \to 0$ as $\Delta x_i \to 0$.


Chain Rule for Real-Valued Functions

Let $f: \R^n \to \R, \mathbf x \mapsto z$ be a differentiable real-valued function. Let $\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \in \R^n$. Further, let every element $x_i: 1 \le i \le n$ represent an implicitly defined differentiable real function of $t$. Then $z$ is itself differentiable {{WRT|Differentiation}} $t$ and: {{begin-eqn}} {{eqn | l = \frac {\d z} {\d t}       | r = \sum_{k \mathop = 1}^n \frac {\partial z }{\partial x_i} \frac {\d x_i} {\d t} }} {{end-eqn}} where $\dfrac {\partial z} {\partial x_i}$ is the partial derivative of $z$ {{WRT|Differentiation}} $x_i$.


Pre-Measure of Finite Stieltjes Function is Pre-Measure

Let $\mathcal J_{ho}$ denote the collection of half-open intervals in $\R$. Let $f: \R \to \R$ be a finite Stieltjes function. Then the pre-measure of $f$, $\mu_f: \mathcal{J}_{ho} \to \overline \R_{\ge 0}$ is a pre-measure. Here, $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.


Pre-Measure of Finite Stieltjes Function Extends to Unique Measure

Let $\mathcal{J}_{ho}$ denote the collection of half-open intervals in $\R$. Let $f: \R \to \R$ be a finite Stieltjes function. Then the pre-measure of $f$, $\mu_f$, extends uniquely to a measure $\mu$ on $\mathcal B \left({\R}\right)$, the Borel $\sigma$-algebra on $\R$. This unique measure $\mu$ is the measure of $f$.


Stieltjes Function of Measure of Finite Stieltjes Function

Let $f: \R \to \R$ be a finite Stieltjes function. Let $\mu_f$ be the measure of $f$. Let $f_{\mu_f}$ be the Stieltjes function of $\mu_f$. Then $f_{\mu_f} = f$.


Measure of Stieltjes Function of Measure

Let $\mu$ be a measure on $\mathcal B \left({\R}\right)$, the Borel $\sigma$-algebra on $\R$. Suppose that for all $n \in \N$, $\mu$ satisfies: :$\mu \left({ \left[{-n \,.\,.\, n}\right) \ }\right) < +\infty$ Let $f_\mu$ be the Stieltjes function of $\mu$. Let $\mu_{f_\mu}$ be the measure of $f_\mu$. Then $\mu_{f_\mu} = \mu$.


Characterization of Measurable Functions

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f: X \to \overline{\R}$ be an extended real-valued function. Then the following are all equivalent: {{begin-eqn}} {{eqn | n=1       | o=       | r=f\) is measurable \( }} {{eqn | n=2       | o=       | r=\forall \alpha \in \R: \left\{ {x \in X: f \left({x}\right) \le \alpha}\right\} \in \Sigma }} {{eqn | n=2'       | o=       | r=\forall \alpha \in \Q: \left\{ {x \in X: f \left({x}\right) \le \alpha}\right\} \in \Sigma }} {{eqn | n=3       | o=       | r=\forall \alpha \in \R: \left\{ {x \in X: f \left({x}\right) < \alpha}\right\} \in \Sigma }} {{eqn | n=3'       | o=       | r=\forall \alpha \in \Q: \left\{ {x \in X: f \left({x}\right) < \alpha}\right\} \in \Sigma }} {{eqn | n=4       | o=       | r=\forall \alpha \in \R: \left\{ {x \in X: f \left({x}\right) \ge \alpha}\right\} \in \Sigma }} {{eqn | n=4'       | o=       | r=\forall \alpha \in \Q: \left\{ {x \in X: f \left({x}\right) \ge \alpha}\right\} \in \Sigma }} {{eqn | n=5       | o=       | r=\forall \alpha \in \R: \left\{ {x \in X: f \left({x}\right) > \alpha}\right\} \in \Sigma }} {{eqn | n=5'       | o=       | r=\forall \alpha \in \Q: \left\{ {x \in X: f \left({x}\right) > \alpha}\right\} \in \Sigma }} {{end-eqn}}


Characterization of Extended Real Sigma-Algebra

Let $\mathcal B \left({\R}\right)$ be the Borel $\sigma$-algebra on $\R$. Let $\overline{\mathcal B}$ be the extended real $\sigma$-algebra. Define $\mathcal S := \mathcal P \left({\left\{{+\infty, -\infty}\right\}}\right)$, where $\mathcal P$ denotes power set. Then: :$\overline{\mathcal B} = \left\{{B \cup S: B \in \mathcal B \left({\R}\right), S \in \mathcal S}\right\}$


Extended Real Sigma-Algebra Induces Borel Sigma-Algebra on Reals

Let $\overline \BB$ be the extended real $\sigma$-algebra. Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$. Then: :$\overline \BB_\R = \map \BB \R$ where $\overline \BB_\R$ denotes a trace $\sigma$-algebra.


Generators for Extended Real Sigma-Algebra

Let $\overline{\mathcal B}$ be the extended real $\sigma$-algebra. Then $\overline{\mathcal B}$ is generated by each of the following collections of extended real intervals: {{begin-eqn}} {{eqn | n = 1       | o =        | r = \left\{ {\ \left[{a \,.\,.\, +\infty}\right]: a \in \R}\right\} }} {{eqn | n = 1'       | o =        | r = \left\{ {\ \left[{a \,.\,.\, +\infty}\right]: a \in \Q}\right\} }} {{eqn | n = 2       | o =        | r = \left\{ {\ \left({b \,.\,.\, +\infty}\right]: b \in \R}\right\} }} {{eqn | n = 2'       | o =        | r = \left\{ {\ \left({b \,.\,.\, +\infty}\right]: b \in \Q}\right\} }} {{eqn | n = 3       | o =        | r = \left\{ {\ \left[{-\infty \,.\,.\, c}\right): c \in \R}\right\} }} {{eqn | n = 3'       | o =        | r = \left\{ {\ \left[{-\infty \,.\,.\, c}\right): c \in \Q}\right\} }} {{eqn | n = 4       | o =        | r = \left\{ {\ \left[{-\infty \,.\,.\, d}\right]: d \in \R}\right\} }} {{eqn | n = 4'       | o =        | r = \left\{ {\ \left[{-\infty \,.\,.\, d}\right]: d \in \Q}\right\} }} {{end-eqn}}


Characteristic Function Measurable iff Set Measurable

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $E \subseteq X$. Then the following are equivalent: :$(1): \quad E \in \Sigma$; i.e., $E$ is a $\Sigma$-measurable set :$(2): \quad \chi_E: X \to \left\{{0, 1}\right\}$, the characteristic function of $E$, is $\Sigma$-measurable


Simple Function is Measurable

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f: X \to \R$ be a simple function. Then $f$ is $\Sigma$-measurable.


Identity Mapping is Relation Isomorphism

Let $\struct {S, \RR}$ be a relational structure. Then the identity mapping $I_S: S \to S$ is a relation isomorphism from $\struct {S, \RR}$ to itself.


Inverse of Relation Isomorphism is Relation Isomorphism

Let $\struct {S, \RR_1}$ and $\struct {T, \RR_2}$ be relational structures. Let $\phi: \struct {S, \RR_1} \to \struct {T, \RR_2}$ be a bijection. Then: :$\phi: \struct {S, \RR_1} \to \struct {T, \RR_2}$ is a relation isomorphism {{iff}}: :$\phi^{-1}: \struct {T, \RR_2} \to \struct {S, \RR_1}$ is also a relation isomorphism.


Measurable Function is Simple Function iff Finite Image Set

Let $\struct {X, \Sigma}$ be a measurable space. Let $f: X \to \R$ be a measurable function. Then $f$ is a simple function {{iff}} its image is finite: :$\card {\Img f} < \infty$


Composite of Relation Isomorphisms is Relation Isomorphism

Let $\struct {S_1, \RR_1}$, $\struct {S_2, \RR_2}$ and $\struct {S_3, \RR_3}$ be relational structures. Let: :$\phi: \struct {S_1, \RR_1} \to \struct {S_2, \RR_2}$ and: :$\psi: \struct {S_2, \RR_2} \to \struct {S_3, \RR_3}$ be relation isomorphisms. Then $\psi \circ \phi: \struct {S_1, \RR_1} \to \struct {S_3, \RR_3}$ is also a relation isomorphism.


Pointwise Sum of Simple Functions is Simple Function

Let $\struct {X, \Sigma}$ be a measurable space. Let $f, g : X \to \R$ be simple functions. Then $f + g: X \to \R, \map {\paren {f + g} } x := \map f x + \map g x$ is also a simple function.


Pointwise Product of Simple Functions is Simple Function

Let $\struct {X, \Sigma}$ be a measurable space. Let $f, g : X \to \R$ be simple functions. Then $f \cdot g: X \to \R, \map {\paren {f \cdot g} } x := \map f x \cdot \map g x$ is also a simple function.


Positive Part of Simple Function is Simple Function

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f: X \to \R$ be a simple function. Then $f^+: X \to \R$, the positive part of $f$, is also a simple function.


Negative Part of Simple Function is Simple Function

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f: X \to \R$ be a simple function. Then $f^-: X \to \R$, the negative part of $f$ is also a simple function.


Difference of Positive and Negative Parts

Let $X$ be a set, and let $f: X \to \overline{\R}$ be an extended real-valued function. Let $f^+$, $f^-: X \to \overline{\R}$ be the positive and negative parts of $f$, respectively. Then $f = f^+ - f^-$.


Sum of Positive and Negative Parts

Let $X$ be a set, and let $f: X \to \overline{\R}$ be an extended real-valued function. Let $f^+, f^-: X \to \overline{\R}$ be the positive and negative parts of $f$, respectively. Then $\left\vert{f}\right\vert = f^+ + f^-$, where $\left\vert{f}\right\vert$ is the absolute value of $f$.


Absolute Value of Simple Function is Simple Function

Let $\struct {X, \Sigma}$ be a measurable space. Let $f: X \to \R$ be a simple function. Then $\size f: X \to \R$, the absolute value of $f$, is also a simple function.


Measurable Function Pointwise Limit of Simple Functions

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f: X \to \overline{\R}$ be a $\Sigma$-measurable function. Then there exists a sequence $\left({f_n}\right)_{n \in \N} \in \mathcal E \left({\Sigma}\right)$ of simple functions, such that: :$\forall x \in X: f \left({x}\right) = \displaystyle \lim_{n \to \infty} f_n \left({x}\right)$ That is, such that $f = \displaystyle \lim_{n \to \infty} f_n$ pointwise. The sequence $\left({f_n}\right)_{n \in \N}$ may furthermore be taken to satisfy: :$\forall n \in \N: \left\vert{f_n}\right\vert \le \left\vert{f}\right\vert$ where $\left\vert{f}\right\vert$ denotes the absolute value of $f$.


Finite Cartesian Product of Non-Empty Sets is Non-Empty

Let $S_1, S_2, \ldots, S_n$ be non-non-empty sets. Then their cartesian product $S_1 \times S_2 \times \cdots \times S_n$ is non-empty.


Equivalent Conditions for Dedekind-Infinite Set

For a set $S$, the following conditions are equivalent: :$(1): \quad$ $S$ is Dedekind-infinite. :$(2): \quad$ $S$ has a countably infinite subset. The above equivalence can be proven in Zermelo-Fraenkel set theory. If the axiom of countable choice is accepted, then it can be proven that the following condition is also equivalent to the above two: :$(3): \quad$ $S$ is infinite.


Relation Isomorphism Preserves Reflexivity

Let $\struct {S, \mathcal R_1}$ and $\struct {T, \mathcal R_2}$ be relational structures. Let $\struct {S, \mathcal R_1}$ and $\struct {T, \mathcal R_2}$ be (relationally) isomorphic. Then $\mathcal R_1$ is a reflexive relation {{iff}} $\mathcal R_2$ is also a reflexive relation.


Preimage of Subset under Composite Mapping

Let $S_1, S_2, S_3$ be sets. Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be mappings. Denote with $g \circ f: S_1 \to S_3$ the composition of $g$ and $f$. Let $S_3' \subseteq S_3$ be a subset of $S_3$. Then: :$\paren {g \circ f}^{-1} \sqbrk {S_3'} = \paren {f^{-1} \circ g^{-1} } \sqbrk {S_3'}$ where $g^{-1} \sqbrk {S_3'}$ denotes the preimage of $S_3'$ under $g$.


Infinite Set has Countably Infinite Subset/Proof 4

If the axiom of countable choice is accepted, then it can be proven that every infinite set has a countably infinite subset.


Set is Infinite iff exist Subsets of all Finite Cardinalities

A set $S$ is infinite {{iff}} for all $n \in \N$, there exists a subset of $S$ whose cardinality is $n$.


Relation Isomorphism Preserves Symmetry

Let $\left({S, \mathcal R_1}\right)$ and $\left({T, \mathcal R_2}\right)$ be relational structures. Let $\left({S, \mathcal R_1}\right)$ and $\left({T, \mathcal R_2}\right)$ be (relationally) isomorphic. Then $\mathcal R_1$ is a symmetric relation {{iff}} $\mathcal R_2$ is also a symmetric relation.


Set Difference with Proper Subset

Let $S$ be a set. Let $T \subsetneq S$ be a proper subset of $S$.   Let $S \setminus T$ denote the set difference between $S$ and $T$. Then: :$S \setminus T \ne \O$ where $\O$ denotes the empty set.


Order Isomorphism iff Strictly Increasing Surjection

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be totally ordered sets. A mapping $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ is an order isomorphism iff: : $(1): \quad \phi$ is a surjection : $(2): \quad \forall x, y \in S: x \mathop{\prec_1} y \implies \phi \left({x}\right) \mathop{\prec_2} \phi \left({y}\right)$


Relation Isomorphism Preserves Transitivity

Let $\struct {S, \RR_1}$ and $\struct {T, \RR_2}$ be relational structures. Let $\struct {S, \RR_1}$ and $\struct {T, \RR_2}$ be (relationally) isomorphic. Then $\RR_1$ is a transitive relation {{iff}} $\RR_2$ is a transitive relation.


Pointwise Supremum of Measurable Functions is Measurable

Let $\left({X, \Sigma}\right)$ be a measurable space, and let $I$ be a countable set. Let $\left({f_i}\right)_{i \in I}$, $f_i: X \to \overline \R$ be an $I$-indexed collection of $\Sigma$-measurable functions. Then the pointwise supremum $\displaystyle \sup_{i \mathop \in I} f_i: X \to \overline \R$ is also $\Sigma$-measurable.


Pointwise Infimum of Measurable Functions is Measurable

Let $\left({X, \Sigma}\right)$ be a measurable space, and let $I$ be a countable set. Let $\left({f_i}\right)_{i \in I}$, $f_i: X \to \overline{\R}$ be an $I$-indexed collection of $\Sigma$-measurable functions. Then the pointwise infimum $\displaystyle \inf_{i \mathop \in I} f_i: X \to \overline{\R}$ is also $\Sigma$-measurable.


Pointwise Upper Limit of Measurable Functions is Measurable

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\left({f_n}\right)_{n \in \N}$, $f_n: X \to \overline{\R}$ be a sequence of $\Sigma$-measurable functions. Then the pointwise upper limit $\displaystyle \limsup_{n \to \infty} f_n: X \to \overline{\R}$ is also $\Sigma$-measurable.


Pointwise Lower Limit of Measurable Functions is Measurable

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\left({f_n}\right)_{n \mathop \in \N}$, $f_n: X \to \overline \R$ be a sequence of $\Sigma$-measurable functions. Then the pointwise lower limit: :$\displaystyle \liminf_{n \mathop \to \infty} f_n: X \to \overline \R$ is also $\Sigma$-measurable.


Pointwise Limit of Measurable Functions is Measurable

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\left({f_n}\right)_{n \in \N}$, $f_n: X \to \overline{\R}$ be a sequence of $\Sigma$-measurable functions. Then the pointwise limit $\displaystyle \lim_{n \to \infty} f_n: X \to \overline{\R}$ is also $\Sigma$-measurable.


Pointwise Sum of Measurable Functions is Measurable

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f, g: X \to \overline{\R}$ be $\Sigma$-measurable functions. Assume that the pointwise sum $f + g: X \to \overline{\R}$ is well-defined. Then $f + g$ is a $\Sigma$-measurable function.


Pointwise Difference of Measurable Functions is Measurable

Let $\struct {X, \Sigma}$ be a measurable space. Let $f, g: X \to \overline \R$ be $\Sigma$-measurable functions. Assume that the pointwise difference $f - g: X \to \overline \R$ is well-defined. Then $f - g$ is a $\Sigma$-measurable function.


Pointwise Product of Measurable Functions is Measurable

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f, g: X \to \overline{\R}$ be $\Sigma$-measurable functions. Then the pointwise product $f \cdot g: X \to \overline{\R}$ is also $\Sigma$-measurable.


Pointwise Maximum of Measurable Functions is Measurable

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f, g: X \to \overline{\R}$ be $\Sigma$-measurable functions. Then the pointwise maximum $\max \left({f, g}\right): X \to \overline{\R}$ is also $\Sigma$-measurable.


Function Measurable iff Positive and Negative Parts Measurable

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f: X \to \overline{\R}$ be an extended real-valued function. Let $f^+, f^-: X \to \overline{\R}$ be the positive and negative parts of $f$. Then $f$ is $\Sigma$-measurable iff both $f^+$ and $f^-$ are $\Sigma$-measurable.


Factorization Lemma/Extended Real-Valued Function

Then an extended real-valued function $g: X \to \overline{\R}$ is $\sigma \left({f}\right)$-measurable {{iff}}: :There exists a $\Sigma$-measurable mapping $\tilde g: Y \to \overline{\R}$ such that $g = \tilde g \circ f$ where: :$\sigma \left({f}\right)$ denotes the $\sigma$-algebra generated by $f$


Factorization Lemma

Let $X$ be a set, and $\left({Y, \Sigma}\right)$ be a measurable space. Let $f: X \to Y$ be a mapping.


Piecewise Combination of Measurable Mappings is Measurable/Binary Case

Let $f, g: X \to X'$ be $\Sigma \, / \, \Sigma'$-measurable mappings. Let $E \in \Sigma$ be a measurable set. Define $h: X \to X'$ by: :$\forall x \in X: \map h x := \begin{cases} \map f x & : \text {if $x \in E$} \\ \map g x & : \text {if $x \notin E$} \end{cases}$ Then $h$ is also a $\Sigma \, / \, \Sigma'$-measurable mapping.


Piecewise Combination of Measurable Mappings is Measurable/General Case

Let $\left({E_n}\right)_{n \in \N} \in \Sigma, \displaystyle \bigcup_{n \mathop \in \N} E_n = X$ be a countable cover of $X$ by $\Sigma$-measurable sets. For each $n \in \N$, let $f_n: E_n \to X'$ be a $\Sigma_{E_n} \, / \, \Sigma'$-measurable mapping. Here, $\Sigma_{E_n}$ is the trace $\sigma$-algebra of $E_n$ in $\Sigma$. Suppose that for every $m, n \in \N$, $f_m$ and $f_n$ satisfy: :$(1): \quad f_m \restriction_{E_m \cap E_n} = f_n \restriction_{E_m \cap E_n}$ that is, $f_m$ and $f_n$ coincide whenever both are defined; here $\restriction$ denotes restriction. Define $f: X \to X'$ by: :$\displaystyle \forall n \in \N, x \in E_n: f \left({x}\right) := f_n \left({x}\right)$ Then $f$ is a $\Sigma \, / \, \Sigma'$-measurable mapping.


Piecewise Combination of Measurable Mappings is Measurable

Let $\struct {X, \Sigma}$ and $\struct {X', \Sigma'}$ be measurable spaces.


Function Simple iff Positive and Negative Parts Simple

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $g: X \to \overline{\R}$ be an extended real-valued function. Then $g$ is a simple function {{iff}} its positive part $g^+$ and negative part $g^-$ are simple functions.


Bounded Measurable Function Uniform Limit of Simple Functions

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f: X \to \overline{\R}$ be a bounded $\Sigma$-measurable function. Then there exists a sequence $\left({f_n}\right)_{n \in \N} \in \mathcal E \left({\Sigma}\right)$ of simple functions, such that: :$\forall \epsilon > 0: \exists n \in \N: \forall x \in X: \left\vert{f \left({x}\right) - f_n \left({x}\right)}\right\vert < \epsilon$ That is, such that $f = \displaystyle \lim_{n \to \infty} f_n$ uniformly. The sequence $\left({f_n}\right)_{n \in \N}$ may furthermore be taken to satisfy: :$\forall n \in \N: \left\vert{f_n}\right\vert \le \left\vert{f}\right\vert$ where $\left\vert{f}\right\vert$ denotes the absolute value of $f$.


Integral of Characteristic Function

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $E \in \Sigma$ be a measurable set, and let $\chi_E: X \to \R$ be its characteristic function. Then $I_\mu \left({\chi_E}\right) = \mu \left({E}\right)$, where $I_\mu \left({\chi_E}\right)$ is the $\mu$-integral of $\chi_E$.


Integral of Positive Simple Function is Positive Homogeneous

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f: X \to \R, f \in \mathcal E^+$ be a positive simple function. Let $\lambda \in \R_{\ge 0}$ be a positive real number. Then $I_\mu \left({\lambda \cdot f}\right) = \lambda \cdot I_\mu \left({f}\right)$, where: :$\lambda \cdot f$ is the pointwise $\lambda$-multiple of $f$ :$I_\mu$ denotes $\mu$-integration This can be summarized by saying that $I_\mu$ is positive homogeneous.


Integral of Positive Simple Function is Additive

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f,g: X \to \R$, $f,g \in \mathcal{E}^+$ be positive simple functions. Then $I_\mu \left({f + g}\right) = I_\mu \left({f}\right) + I_\mu \left({g}\right)$, where: :$f + g$ is the pointwise sum of $f$ and $g$ :$I_\mu$ denotes $\mu$-integration This can be summarized by saying that $I_\mu$ is additive.


Strict Ordering Preserved under Product with Invertible Element

Let $\left({S, \circ, \preceq}\right)$ be an ordered semigroup. Let $z \in S$ be invertible. Suppose that either $x \circ z \prec y \circ z$ or $z \circ x \prec z \circ y$. Then $x \prec y$.


Strict Ordering Preserved under Cancellability in Totally Ordered Semigroup

Let $\left({S, \circ, \preceq}\right)$ be a totally ordered semigroup. If either: : $x \circ z \prec y \circ z$ or : $z \circ x \prec z \circ y$ then $x \prec y$.


Relation Compatibility in Totally Ordered Semigroup

Let $\left({S, \circ, \preceq}\right)$ be an ordered semigroup such that: :$(1): \quad$ All the elements of $\left({S, \circ, \preceq}\right)$ are cancellable for $\circ$ :$(2): \quad \preceq$ is a total ordering. Then: :$\forall x, y, z \in S: x \circ z \preceq y \circ z \iff x \preceq y$


Integral of Positive Measurable Function Extends Integral of Positive Simple Function

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f: X \to \R, f \in \mathcal{E}^+$ be a positive simple function. Then $\displaystyle \int f \, \mathrm d\mu = I_\mu \left({f}\right)$, where: :$\displaystyle \int \cdot \, \mathrm d\mu$ denotes the $\mu$-integral of positive measurable functions :$I_\mu$ denotes the $\mu$-integral of positive simple functions That is, $\displaystyle \int \cdot \, \mathrm d\mu \restriction_{\mathcal{E}^+} = I_\mu$, using the notion of restriction, $\restriction$.


Beppo Levi's Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $\sequence {f_n}_{n \mathop \in \N} \in \MM_{\overline \R}^+$ be an increasing sequence of positive $\Sigma$-measurable functions. Let $\displaystyle \sup_{n \mathop \in \N} f_n: X \to \overline \R$ be the pointwise supremum of $\sequence {f_n}_{n \mathop \in \N}$, where $\overline \R$ denotes the extended real numbers. Then: :$\displaystyle \int \sup_{n \mathop \in \N} f_n \rd \mu = \sup_{n \mathop \in \N} \int f_n \rd \mu$ where the supremum on the {{RHS}} is in the ordering on $\overline \R$.


Ordered Semigroup Isomorphism is Surjective Monomorphism

Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered semigroups. Let $\phi: \left({S, \circ, \preceq}\right) \to \left({T, *, \preccurlyeq}\right)$ be a mapping. Then $\phi$ is an ordered semigroup isomorphism iff: :$(1): \quad \phi$ is an ordered semigroup monomorphism :$(2): \quad \phi$ is a surjection.


Ordered Semigroup Monomorphism into Image is Isomorphism

Let $\struct {S, \circ, \preceq}$ and $\struct {T, *, \preccurlyeq}$ be ordered semigroups. Let $\phi: \struct {S, \circ, \preceq} \to \struct {T, *, \preccurlyeq}$ be an ordered semigroup monomorphism. Let $S'$ be the image of $\phi$. Then $\phi$ is an ordered semigroup isomorphism from $\struct {S, \circ, \preceq}$ into $\struct {S', * {\restriction_{S'} }, \preccurlyeq \restriction_{S'} }$. Here: :$* {\restriction_{S'}}$ denotes the restriction of $*$ to $S' \times S'$ :$\preccurlyeq \restriction_{S'}$ denotes the restriction of $\preccurlyeq$ to $S' \times S'$.


Order Completion Unique up to Isomorphism

Let $\left({S, \preceq_S}\right)$ be an ordered set. Suppose that both $\left({T, \preceq_T}\right)$ and $\left({T', \preceq_{T'}}\right)$ are order completions for $\left({S, \preceq_S}\right)$. Then there exists a unique order isomorphism $\psi: T \to T'$. In particular, $\left({T, \preceq_T}\right)$ and $\left({T', \preceq_{T'}}\right)$ are isomorphic.


Intersection of Strict Upper Closures in Toset

Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $a, b \in S$. Then: :$a^\succ \cap b^\succ = \left({\max \left({a, b}\right)}\right)^\succ$ where: :$a^\succ$ denotes strict upper closure of $a$ :$\max$ denotes the max operation.


Intersection of Weak Upper Closures in Toset

Let $\left({S, \preccurlyeq}\right)$ be a totally ordered set. Let $a, b \in S$. Then: :$a^\succcurlyeq \cap b^\succcurlyeq = \left({\max \left({a, b}\right)}\right)^\succcurlyeq$ where: : $a^\succcurlyeq$ denotes weak upper closure of $a$ : $\max$ denotes the max operation.


Intersection of Weak Lower Closures in Toset

Let $\struct {S, \preccurlyeq}$ be a totally ordered set. Let $a, b \in S$. Then: :$a^\preccurlyeq \cap b^\preccurlyeq = \paren {\min \set {a, b} }^\preccurlyeq$ where: :$a^\preccurlyeq$ denotes the weak lower closure of $a$ :$\min$ denotes the min operation.


Naturally Ordered Semigroup Exists

There exists a Naturally Ordered Semigroup.


Strict Upper Closure in Restricted Ordering

Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$ be a subset of $S$, and let $\preceq \restriction_T$ be the restricted ordering on $T$. Then for all $t \in T$: :$t^{\succ T} = T \cap t^{\succ S}$ where: : $t^{\succ T}$ is the strict upper closure of $t$ in $\left({T, \preceq \restriction_T}\right)$ : $t^{\succ S}$ is the strict upper closure of $t$ in $\left({S, \preceq}\right)$.


Weak Upper Closure in Restricted Ordering

Let $\left({S, \preccurlyeq}\right)$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. Let $\preccurlyeq \restriction_T$ be the restricted ordering on $T$. Then for all $t \in T$: :$t^{\succcurlyeq T} = T \cap t^{\succcurlyeq S}$ where: : $t^{\succcurlyeq T}$ is the weak upper closure of $t$ in $\left({T, \preccurlyeq \restriction_T}\right)$ : $t^{\succcurlyeq S}$ is the weak upper closure of $t$ in $\left({S, \preccurlyeq}\right)$.


Strict Lower Closure in Restricted Ordering

Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$ be a subset of $S$, and let $\preceq \restriction_T$ be the restricted ordering on $T$. Then for all $t \in T$: :$t^{\prec T} = T \cap t^{\prec S}$ where: : $t^{\prec T}$ is the strict lower closure of $t$ in $\left({T, \preceq \restriction_T}\right)$ : $t^{\prec S}$ is the strict lower closure of $t$ in $\left({S, \preceq}\right)$.


Weak Lower Closure in Restricted Ordering

Let $\left({S, \preccurlyeq}\right)$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. Let $\preccurlyeq \restriction_T$ be the restricted ordering on $T$. Then for all $t \in T$: :$t^{\preccurlyeq T} = T \cap t^{\preccurlyeq S}$ where: :$t^{\preccurlyeq T}$ is the weak lower closure of $t$ in $\left({T, \preccurlyeq \restriction_T}\right)$ :$t^{\preccurlyeq S}$ is the weak lower closure of $t$ in $\left({S, \preccurlyeq}\right)$.


Natural Numbers under Multiplication form Ordered Commutative Semigroup

Let $\N$ be the natural numbers. Let $\times$ be multiplication. Let $\le$ be the ordering on $\N$. Then $\left({\N, \times, \le}\right)$ is an ordered commutative semigroup.


Invertible Elements under Natural Number Multiplication

Let $\N$ be the natural numbers. Let $\times$ denote multiplication. Then the only invertible element of $\N$ for $\times$ is $1$.


Graph containing Closed Walk of Odd Length also contains Odd Cycle

Let $G$ be a graph. {{explain|This proof works for a simple graph, but the theorem may hold for loop graphs and/or multigraphs. Clarification needed as to what applies.}} Let $G$ have a closed walk of odd length. Then $G$ has an odd cycle.


Homomorphism of Powers/Naturally Ordered Semigroup

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. For a given $a \in T_1$, let $\map {\odot^n} a$ be the $n$th power of $a$ in $T_1$. For a given $a \in T_2$, let $\map {\oplus^n} a$ be the $n$th power of $a$ in $T_2$. Then: :$\forall a \in T_1: \forall n \in \struct {S^*, \circ, \preceq}: \map \phi {\map {\odot^n} a} = \map {\oplus^n} {\map \phi a}$ where $S^* = S \setminus \set 0$.


Homomorphism of Powers/Natural Numbers

Let $n \in \N$. Let $\odot^n$ and $\oplus^n$ be the $n$th powers of $\odot$ and $\oplus$, respectively. Then: :$\forall a \in T_1: \forall n \in \N: \map \phi {\map {\odot^n} a} = \map {\oplus^n} {\map \phi a}$


Homomorphism of Powers/Integers

Let $\struct {T_1, \odot}$ and $\struct {T_2, \oplus}$ be monoids. Let $\phi: \struct {T_1, \odot} \to \struct {T_2, \oplus}$ be a (semigroup) homomorphism. Let $a$ be an invertible element of $T_1$. Let $n \in \Z$. Let $\odot^n$ and $\oplus^n$ be as defined as in Index Laws for Monoids. Then: :$\forall n \in \Z: \map \phi {\map {\odot^n} a} = \map {\oplus^n} {\map \phi a}$


Right Operation is Left Distributive over All Operations

Let $\struct {S, \circ, \rightarrow}$ be an algebraic structure where: :$\rightarrow$ is the right operation :$\circ$ is any arbitrary binary operation. Then $\rightarrow$ is left distributive over $\circ$.


Left Operation is Right Distributive over All Operations

Let $\struct {S, \circ, \leftarrow}$ be an algebraic structure where: :$\leftarrow$ is the left operation :$\circ$ is any arbitrary binary operation. Then $\leftarrow$ is right distributive over $\circ$.


Right Operation is Distributive over Idempotent Operation

Let $\struct {S, \circ, \rightarrow}$ be an algebraic structure where: :$\rightarrow$ is the right operation :$\circ$ is any arbitrary binary operation. Then: :$\rightarrow$ is distributive over $\circ$ {{iff}} :$\circ$ is idempotent.


Left Operation is Distributive over Idempotent Operation

Let $\struct {S, \circ, \leftarrow}$ be an algebraic structure where: :$\leftarrow$ is the left operation :$\circ$ is any arbitrary binary operation. Then: :$\leftarrow$ is distributive over $\circ$ {{iff}} :$\circ$ is idempotent.


Integral of Positive Measurable Function as Limit of Integrals of Positive Simple Functions

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f: X \to \overline{\R} \in \mathcal{M}_{\overline{\R}}^+$ be a positive $\Sigma$-measurable function. Let $\left({f_n}\right)_{n \in \N} \in \mathcal{E}^+$, $f_n: X \to \R$ be a sequence of positive simple functions such that: :$\displaystyle \lim_{n \to \infty} f_n = f$ where $\lim$ denotes a pointwise limit. Then: :$\displaystyle \int f \, \mathrm d\mu = \lim_{n \to \infty} \int f_n \, \mathrm d\mu$ where the integral signs denote $\mu$-integration.


Integral of Characteristic Function/Corollary

:$\displaystyle \int \chi_E \, \mathrm d\mu = \mu \left({E}\right)$ where the integral sign denotes the $\mu$-integral of $\chi_E$.


Integral of Positive Measurable Function is Positive Homogeneous

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f: X \to \R, f \in \mathcal M_{\overline \R}^+$ be a positive measurable function. Let $\lambda \in \R_{\ge 0}$ be a positive real number. Then: :$\displaystyle \int \lambda f \rd \mu = \lambda \int f \rd \mu$ where: :$\lambda f$ is the pointwise $\lambda$-multiple of $f$ :The integral sign denotes $\mu$-integration This can be summarized by saying that $\displaystyle \int \cdot \rd \mu$ is positive homogeneous.


Linear Transformation as Matrix Product

Let $T: \R^n \to \R^m, \mathbf x \mapsto \map T {\mathbf x}$ be a linear transformation. Then: :$\map T {\mathbf x} = \mathbf A_T \mathbf x$ where $\mathbf A_T$ is the $m \times n$ matrix defined as: :$\mathbf A_T = \begin {bmatrix} \map T {\mathbf e_1} & \map T {\mathbf e_2} & \cdots & \map T {\mathbf e_n} \end {bmatrix}$ where $\tuple {\mathbf e_1, \mathbf e_2, \cdots, \mathbf e_n}$ is the standard ordered basis of $\R^n$.


Integral of Positive Measurable Function is Additive

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f,g: X \to \overline{\R}$, $f,g \in \mathcal{M}_{\overline{\R}}^+$ be positive measurable functions. Then: :$\displaystyle \int f + g \, \mathrm d\mu = \displaystyle \int f \, \mathrm d\mu + \displaystyle \int g \, \mathrm d\mu$ where: :$f + g$ is the pointwise sum of $f$ and $g$ :The integral sign denotes $\mu$-integration This can be summarized by saying that $\displaystyle \int \cdot \, \mathrm d\mu$ is additive.


Integral of Positive Simple Function is Increasing

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f, g: X \to \R$, $f, g \in \mathcal{E}^+$ be positive simple functions. Suppose that: : $f \le g$ where $\le$ denotes pointwise inequality. Then: :$I_\mu \left({f}\right) \le I_\mu \left({g}\right)$ where $I_\mu$ denotes $\mu$-integration This can be summarized by saying that $I_\mu$ is an increasing mapping.


Integral of Positive Measurable Function is Monotone

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f,g: X \to \overline{\R}$, $f,g \in \mathcal{M}_{\overline{\R}}^+$ be positive measurable functions. Suppose that $f \le g$, where $\le$ denotes pointwise inequality. Then: :$\displaystyle \int f \, \mathrm d\mu \le \int g \, \mathrm d\mu$ where the integral sign denotes $\mu$-integration. This can be summarized by saying that $\displaystyle \int \cdot \, \mathrm d\mu$ is monotone.


Matrix Multiplication is Homogeneous of Degree 1

Let $\mathbf A$ be an $m \times n$ matrix and $\mathbf B$ be an $n \times p$ matrix such that the columns of $\mathbf A$ and $\mathbf B$ are members of $\R^m$ and $\R^n$, respectively. Let $\lambda \in \mathbb F \in \set {\R, \C}$ be a scalar. Then: :$\mathbf A \paren {\lambda \mathbf B} = \lambda \paren {\mathbf A \mathbf B}$


Series of Positive Measurable Functions is Positive Measurable Function

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\left({f_n}\right)_{n \in \N} \in \mathcal{M}_{\overline{\R}}^+$, $f_n: X \to \overline{\R}$ be a sequence of positive measurable functions. Let $\displaystyle \sum_{n \mathop \in \N} f_n: X \to \overline{\R}$ be the pointwise series of the $f_n$. Then $\displaystyle \sum_{n \mathop \in \N} f_n$ is also a positive measurable function.


Integral of Series of Positive Measurable Functions

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\left({f_n}\right)_{n \in \N} \in \mathcal{M}_{\overline{\R}}^+$, $f_n: X \to \overline{\R}$ be a sequence of positive measurable functions. Let $\displaystyle \sum_{n \mathop \in \N} f_n: X \to \overline{\R}$ be the pointwise series of the $f_n$. Then: :$\displaystyle \int \sum_{n \mathop \in \N} f_n \, \mathrm d \mu = \sum_{n \mathop \in \N} \int f_n \, \mathrm d \mu$ where the integral sign denotes $\mu$-integration.


Integral with respect to Discrete Measure

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\displaystyle \mu = \sum_{n \mathop \in \N} \lambda_n \delta_{x_n}$ be a discrete measure on $\left({X, \Sigma}\right)$. Let $f \in \mathcal{M}_{\overline{\R}}^+, f: X \to \overline{\R}$ be a positive measurable function. Then: :$\displaystyle \int f \, \mathrm d\mu = \sum_{n \mathop \in \N} \lambda_n f \left({x_n}\right)$ where the integral sign denotes $\mu$-integration.


Matrix Product as Linear Transformation

Let: :$ \mathbf A_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$ :$\mathbf x_{n \times 1} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$ :$\mathbf y_{n \times 1} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}$ be matrices where each column is an element of a real vector space. Let $T$ be the mapping: :$T: \R^m \to \R^n, \mathbf x \mapsto \mathbf A \mathbf x$ Then $T$ is a linear transformation.


Fatou's Lemma for Integrals

Let $\struct {X, \Sigma, \mu}$ be a measure space.


Linear Transformation Maps Zero Vector to Zero Vector

Let $\mathbf V$ be a vector space, with zero $\mathbf 0$. Likewise let $\mathbf V\,'$ be another vector space, with zero $\mathbf 0'$. Let $T: \mathbf V \to \mathbf V\,'$ be a linear transformation. Then: :$T: \mathbf 0 \mapsto \mathbf 0'$


Infimum of Product

Let $\left({G, \circ, \preceq}\right)$ be an ordered group. Suppose that subsets $A$ and $B$ of $G$ admit infima in $G$. Then: :$\inf \left({A \circ_{\mathcal P} B}\right) = \inf A \circ \inf B$ where $\circ_{\mathcal P}$ denotes subset product.


Supremum of Product

Let $\struct {G, \circ, \preceq}$ be an ordered group. Suppose that subsets $A$ and $B$ of $G$ admit suprema in $G$. Then: :$\sup \paren {A \circ_{\PP} B} = \sup A \circ \sup B$ where $\circ_{\PP}$ denotes subset product.


Integral with respect to Series of Measures

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\displaystyle \mu := \sum_{n \mathop \in \N} \lambda_n \mu_n$ be a series of measures on $\left({X, \Sigma}\right)$. Then for all positive measurable functions $f: X \to \overline \R, f \in \mathcal M_{\overline{\R}}^+$: :$\displaystyle \int f \, \mathrm d \mu = \sum_{n \mathop \in \N} \int f \, \mathrm d \mu_n$ where the integral signs denote integration with respect to a measure.


Reverse Fatou's Lemma


Characteristic Function of Limit Inferior of Sequence of Sets

Let $\left({E_n}\right)_{n \in \N}$ be a sequence of sets. Let $E := \displaystyle \liminf_{n \mathop \to \infty} \, E_n$ be the limit inferior of the $E_n$. Then: :$\displaystyle \chi_E = \liminf_{n \mathop \to \infty} \, \chi_{E_n}$ where: :$\chi$ denotes characteristic function :$\displaystyle \liminf_{n \to \infty} \, \chi_{E_n}$ is the pointwise limit inferior of the $\chi_{E_n}$


Characteristic Function of Limit Superior of Sequence of Sets

Let $\left({E_n}\right)_{n \in \N}$ be a sequence of sets. Let $E := \displaystyle \limsup_{n \mathop \to \infty} \, E_n$ be the limit superior of the $E_n$. Then: :$\displaystyle \chi_E = \limsup_{n \to \infty} \, \chi_{E_n}$ where: :$\chi$ denotes characteristic function :$\displaystyle \liminf_{n \to \infty} \, \chi_{E_n}$ is the pointwise limit superior of the $\chi_{E_n}$


Fatou's Lemma for Measures

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $\sequence {E_n}_{n \mathop \in \N} \in \Sigma$ be a sequence of $\Sigma$-measurable sets. Then: :$\displaystyle \map \mu {\liminf_{n \mathop \to \infty} E_n} \le \liminf_{n \mathop \to \infty} \map \mu {E_n}$ where: :$\displaystyle \liminf_{n \mathop \to \infty} E_n$ is the limit inferior of the $E_n$ :the {{RHS}} limit inferior is taken in the extended real numbers $\overline \R$.


Kernel Transformation of Measure is Measure

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $N: X \times \Sigma \to \overline{\R}_{\ge0}$ be a kernel. Then $\mu N: X \to \overline{\R}$, the kernel transformation of $\mu$, is a measure.


Kernel Transformation of Positive Measurable Function is Positive Measurable Function

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $N: X \times \Sigma \to \overline{\R}_{\ge0}$ be a kernel. Let $f: X \to \overline{\R}$ be a positive measurable function. Then $N f: X \to \overline{\R}$, the transformation of $f$ by $N$, is also a positive measurable function.


Integral with respect to Kernel Transformation of Measure

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $N: X \times \Sigma \to \overline \R_{\ge 0}$ be a kernel. Let $f: X \to \overline \R$ be a positive measurable function. Then: :$\displaystyle \int f \map \rd {\mu N} = \int N f \rd \mu$ where: :The integral sign denotes integration with respect to a measure :$\mu N$ is the transformation of $\mu$ by $N$ :$N f$ is the transformation of $f$ by $N$ Writing $\map \mu f$ in place of $\displaystyle \int f \rd \mu$, the theorem statement can be conveniently expressed as: :$\map {\mu N} f = \map \mu {N f}$


Canonical Injection is Injection

Let $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ be algebraic structures with identities $e_1, e_2$ respectively. The canonical injections: :$\inj_1: \struct {S_1, \circ_1} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_1: \map {\inj_1} x = \tuple {x, e_2}$ :$\inj_2: \struct {S_2, \circ_2} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_2: \map {\inj_2} x = \tuple {e_1, x}$ are injections.


Canonical Injection is Injection/General Result

Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \dotsc, \struct {S_j, \circ_j}, \dotsc, \struct {S_n, \circ_n}$ be algebraic structures with identities $e_1, e_2, \ldots, e_j, \ldots, e_n$ respectively. The canonical injection: :$\displaystyle \inj_j: \struct {S_j, \circ_j} \to \prod_{i \mathop = 1}^n \struct {S_i, \circ_i}$ defined as: :$\map {\inj_j} x = \tuple {e_1, e_2, \dotsc, e_{j - 1}, x, e_{j + 1}, \dotsc, e_n}$ is an injection.


Finite Union of Countable Sets is Countable

The union of a finite number of countable sets is countable.


Particular Values of Binomial Coefficients

=== Binomial Coefficient $\dbinom 0 0$ === {{:Zero Choose Zero}} === Binomial Coefficient $\dbinom 0 n$ === {{:Zero Choose n}} === Binomial Coefficient $\dbinom 1 n$ === {{:One Choose n}} === N Choose Negative Number is Zero === {{:N Choose Negative Number is Zero}} === Binomial Coefficient with Zero === {{:Binomial Coefficient with Zero}} === Binomial Coefficient with One === {{:Binomial Coefficient with One}} === Binomial Coefficient with Self === {{:Binomial Coefficient with Self}} === Binomial Coefficient with Self minus One === {{:Binomial Coefficient with Self minus One}} === Binomial Coefficient with Two === {{:Binomial Coefficient with Two}}


Binomial Coefficient with Zero

:$\forall r \in \R: \dbinom r 0 = 1$


Binomial Coefficient with One

:$\forall r \in \R: \dbinom r 1 = r$


Binomial Coefficient with Self

:$\forall n \in \Z: \dbinom n n = \sqbrk {n \ge 0}$ where $\sqbrk {n \ge 0}$ denotes Iverson's convention. That is: :$\forall n \in \Z_{\ge 0}: \dbinom n n = 1$ :$\forall n \in \Z_{< 0}: \dbinom n n = 0$


Binomial Coefficient with Two

:$\forall r \in \R: \dbinom r 2 = \dfrac {r \left({r - 1}\right)} 2$


Binomial Theorem/Integral Index

Let $X$ be one of the set of numbers $\N, \Z, \Q, \R, \C$. Let $x, y \in X$. Then: {{begin-eqn}} {{eqn | ll= \forall n \in \Z_{\ge 0}:       | l = \paren {x + y}^n       | r = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k       | c =  }} {{eqn | r = x^n + \binom n 1 x^{n - 1} y + \binom n 2 x^{n - 2} y^2 + \binom n 3 x^{n - 3} y^3 + \cdots       | c =  }} {{eqn | r = x^n + n x^{n - 1} y + \frac {n \paren {n - 1} } {2!} x^{n - 2} y^2 + \frac {n \paren {n - 1} \paren {n - 3} } {3!} x^{n - 3} y^3 + \cdots       | c =  }} {{end-eqn}} where $\dbinom n k$ is $n$ choose $k$.


Binomial Theorem/Ring Theory

Let $\left({R, +, \odot}\right)$ be a ringoid such that $\left({R, \odot}\right)$ is a commutative semigroup. Let $n \in \Z: n \ge 2$. Then: :$\displaystyle \forall x, y \in R: \odot^n \left({x + y}\right) = \odot^n x + \sum_{k \mathop = 1}^{n-1} \binom n k \left({\odot^{n-k} x}\right) \odot \left({\odot^k y}\right) + \odot^n y$ where $\dbinom n k = \dfrac {n!} {k! \ \left({n - k}\right)!}$ (see Binomial Coefficient). If $\left({R, \odot}\right)$ has an identity element $e$, then: :$\displaystyle \forall x, y \in R: \odot^n \left({x + y}\right) = \sum_{k \mathop = 0}^n \binom n k \left({\odot^{n - k} x}\right) \odot \left({\odot^k y}\right)$


Derivative of Absolute Value Function

Let $\size x$ be the absolute value of $x$ for real $x$. Then: :$\dfrac \d {\d x} \size x = \dfrac x {\size x}$ for $x \ne 0$. At $x = 0$, $\size x$ is not differentiable.


Inverse Element in Inverse Completion of Commutative Monoid

Let $\left({S, \circ}\right)$ be a commutative monoid. Let $\left ({C, \circ}\right) \subseteq \left({S, \circ}\right)$ be the subsemigroup of cancellable elements of $\left({S, \circ}\right)$. Let $\left({T, \circ'}\right)$ be an inverse completion of $\left({S, \circ}\right)$. Then the inverse of an element of $S$ which is invertible for $\circ$ is also its inverse for $\circ'$.


Construction of Inverse Completion/Congruence Relation

The cross-relation $\boxtimes$ is a congruence relation on $\left({S \times C, \oplus}\right)$. === Members of Equivalence Classes === {{:Construction of Inverse Completion/Equivalence Relation/Members of Equivalence Classes}} === Equivalence Class of Equal Elements === {{:Construction of Inverse Completion/Equivalence Relation/Equivalence Class of Equal Elements}}


Construction of Inverse Completion/Equivalence Relation/Members of Equivalence Classes

$\forall x, y \in S, a, b \in C:$ :$(1): \quad \tuple {x \circ a, a} \boxtimes \tuple {y \circ b, b} \iff x = y$ :$(2): \quad \eqclass {\tuple {x \circ a, y \circ a} } \boxtimes = \eqclass {\tuple {x, y} } \boxtimes$ where $\eqclass {\tuple {x, y} } \boxtimes$ is the equivalence class of $\tuple {x, y}$ under $\boxtimes$.


Construction of Inverse Completion/Equivalence Relation/Equivalence Class of Equal Elements

:$\forall c, d \in C: \left({c, c}\right) \boxtimes \left({d, d}\right)$


Construction of Inverse Completion/Quotient Structure is Commutative Semigroup

:$\left({T', \oplus'}\right)$ is a commutative semigroup.


Construction of Inverse Completion/Quotient Mapping is Injective

Let the mapping $\psi: S \to T'$ be defined as: :$\forall x \in S: \psi \left({x}\right) = \left[\!\left[{\left({x \circ a, a}\right)}\right]\!\right]_\boxtimes$ Then $\psi: S \to T'$ is an injection, and does not depend on the particular element $a$ chosen.


Construction of Inverse Completion/Quotient Mapping is Monomorphism

The mapping $\psi: S \to T'$ is a monomorphism.


Construction of Inverse Completion/Quotient Mapping to Image is Isomorphism

Let $S'$ be the image $\psi \left({S}\right)$ of $S$. Then $\psi$ is an isomorphism from $S$ onto $S'$.


Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements

The set $C'$ of cancellable elements of the semigroup $S'$ is $\psi \left[{C}\right]$.