hcju/proofwikips huggingface dataset:
Definition:Half-Open Rectangle
Let $a_1, \ldots, a_n, b_1, \ldots, b_n$ be real numbers. The set: :$\ds \prod_{i \mathop = 1}^n \hointr {a_i} {b_i} = \hointr {a_1} {b_1} \times \cdots \times \hointr {a_n} {b_n} \subseteq \R^n$ is called an '''half-open rectangle in $\R^n$''' or '''half-open $n$-rectangle'''. Here, $\times$ denotes Cartesian product. The collection of all '''half-open $n$-rectangles''' is denoted $\JJ_{ho}$, or $\JJ_{ho}^n$ if the dimension $n$ is to be emphasized. In case $a_i \ge b_i$ for some $i$, the rectangle is taken to be the empty set $\O$. This is in accordance with the result Cartesian Product is Empty iff Factor is Empty for general Cartesian products.
Definition:Discrete Measure
Let $\struct {X, \Sigma, \mu}$ be a measure space. Then $\mu$ is said to be a '''discrete measure'''  it is a series of Dirac measures. That is,  there exist: :A sequence $\sequence {x_n}_{n \mathop \in \N}$ in $X$ :A sequence $\sequence {\lambda_n}_{n \mathop \in \N}$ in $\R$ such that: :$(1):\quad \forall E \in \Sigma: \map \mu E = \displaystyle \sum_{n \mathop \in \N} \lambda_n \, \map {\delta_{x_n} } E$ where $\delta_{x_n}$ denotes the Dirac measure at $x_n$. By Series of Measures is Measure, defining $\mu$ by $(1)$ yields a measure.
Definition:Series of Measures
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\left({\mu_n}\right)_{n \in \N}$ be a sequence of measures on $\left({X, \Sigma}\right)$. Let $\left({\lambda_n}\right)_{n \in \N}$ be a sequence of positive real numbers. Then the mapping $\mu: \Sigma \to \overline \R$, defined by: :$\displaystyle \mu \left({E}\right) := \sum_{n \mathop \in \N} \lambda_n \mu_n \left({E}\right)$ is called a '''series of measures'''.
Definition:Induced Outer Measure
Let $\mathcal S$ be a collection of subsets of a set $X$, and suppose that $\varnothing \in \mathcal S$. Let $\mu$ be a pre-measure on $\mathcal S$. The '''outer measure induced by the pre-measure $\mu$''' is the mapping $\mu^* : \mathcal P \left({X}\right) \to \overline \R_{\ge 0}$ defined as: :$\displaystyle \mu^* \left({S}\right) = \inf \ \left\{ {\sum_{n \mathop = 1}^\infty \mu \left({A_n}\right) : \forall n \in \N: A_n \in \mathcal S, \ S \subseteq \bigcup_{n \mathop = 1}^\infty A_n} \right\}$ Here, $\mathcal P \left({X}\right)$ denotes the power set of $X$, and $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers. The infimum of the empty set is taken to be $+\infty$. It follows immediately by Construction of Outer Measure that the induced outer measure is an outer measure.
Definition:Homomorphism (Abstract Algebra)/Cartesian Product
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from one algebraic structure $\struct {S, \circ}$ to another $\struct {T, *}$. We define the cartesian product $\phi \times \phi: S \times S \to T \times T$ as: :$\forall \tuple {x, y} \in S \times S: \map {\paren {\phi \times \phi} } {x, y} = \tuple {\map \phi x, \map \phi y}$ Hence we can state that $\phi$ is a homomorphism : :$\map \ast {\map {\paren {\phi \times \phi} } {x, y} } = \map \phi {\map \circ {x, y} }$ using the notation $\map \circ {x, y}$ to denote the operation $x \circ y$. The point of doing this is so we can illustrate what is going on in a commutative diagram: ::$\begin{xy} \xymatrix@L+2mu@+1em{  S \times S \ar[r]^*{\circ}      \ar[d]_*{\phi \times \phi} &  S \ar[d]^*{\phi} \\  T \times T  \ar[r]_*{\ast} &  T }\end{xy}$ Thus we see that $\phi$ is a homomorphism  both of the composite mappings from $S \times S$ to $T$ have the same effect on all elements of $S \times S$.
Definition:Epimorphism (Category Theory)
Let $\mathbf C$ be a metacategory. An '''epimorphism''' is a morphism $f \in \mathbf C_1$ such that: : $g \circ f = h \circ f \implies g = h$ for all morphisms $g, h \in \mathbf C_1$ for which these compositions are defined. That is, an '''epimorphism''' is a morphism which is right cancellable.  One writes $f: C \twoheadrightarrow D$ to denote that $f$ is an '''epimorphism'''.
Definition:Ordered Structure Monomorphism
Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered structures. An '''ordered structure monomorphism''' from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ A monomorphism, i.e. an injective homomorphism, from the structure $\left({S, \circ}\right)$ to the structure $\left({T, *}\right)$ :$(2): \quad$ An order embedding from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.
Definition:Monomorphism (Category Theory)
Let $\mathbf C$ be a metacategory. A '''monomorphism''' is a morphism $f \in \mathbf C_1$ such that: :$f \circ g = f \circ h \implies g = h$ for all morphisms $g, h \in \mathbf C_1$ for which these compositions are defined. That is, a '''monomorphism''' is a morphism which is left cancellable.  One writes $f: C \rightarrowtail D$ to denote that $f$ is a '''monomorphism'''.
Definition:Complex Number/Real Part
Let $z = a + i b$ be a complex number. The '''real part''' of $z$ is the coefficient $a$. The '''real part''' of a complex number $z$ is usually denoted on  by $\map \Re z$ or $\Re z$.
Definition:Complex Number/Imaginary Part
Let $z = a + i b$ be a complex number. The '''imaginary part''' of $z$ is the coefficient $b$ ('''note:''' not $i b$). The '''imaginary part''' of a complex number $z$ is usually denoted on  by $\map \Im z$ or $\Im z$.
Definition:Strict Lower Closure
=== Strict Lower Closure of Element === {{:Definition:Strict Lower Closure/Element}} === Strict Lower Closure of Subset === {{:Definition:Strict Lower Closure/Set}}
Definition:Strict Upper Closure
=== Strict Upper Closure of Element === {{:Definition:Strict Upper Closure/Element}} === Strict Upper Closure of Subset === {{:Definition:Strict Upper Closure/Set}}
Definition:Upper Closure/Element
Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $a \in S$. The '''upper closure of $a$ (in $S$)''' is defined as: :$a^\succcurlyeq := \set {b \in S: a \preccurlyeq b}$ That is, $a^\succcurlyeq$ is the set of all elements of $S$ that succeed $a$.
Definition:Order Topology
=== Definition 1 === {{:Definition:Order Topology/Definition 1}} === Definition 2 === {{:Definition:Order Topology/Definition 2}}
Definition:Upper Closure
{{:Definition:Upper Closure/Element}}
Definition:Lower Closure
{{:Definition:Lower Closure/Element}}
Definition:Projection (Mapping Theory)/First Projection
The '''first projection on $S \times T$''' is the mapping $\pr_1: S \times T \to S$ defined by: :$\forall \tuple {x, y} \in S \times T: \map {\pr_1} {x, y} = x$
Definition:Projection (Mapping Theory)/Second Projection
The '''second projection on $S \times T$''' is the mapping $\pr_2: S \times T \to T$ defined by: :$\forall \tuple {x, y} \in S \times T: \map {\pr_2} {x, y} = y$
Definition:Ordering on Extended Real Numbers
Let $\overline \R$ denote the extended real numbers. Extend the natural ordering $\le_\R$ on $\R$ to $\overline \R = \R \cup \set {+\infty, -\infty}$ by imposing: :$\forall x \in \overline \R: -\infty \le x$ :$\forall x \in \overline \R: x \le +\infty$ That is, considering the relations $\le$ and $\le_\R$ as subsets of $\overline \R \times \overline \R$: :${\le} := {\le_\R} \cup \set {\tuple {x, +\infty}: x \in \overline \R} \cup \set {\tuple {-\infty, x}: x \in \overline \R}$ where $\tuple {x, +\infty}$ and $\tuple {-\infty, x}$ denote ordered pairs in $\overline \R \times \overline \R$. The ordering $\le$ is called the '''(usual) ordering on $\overline \R$'''.
Definition:Topology on Extended Real Numbers
Let $\overline \R$ denote the extended real numbers. The '''(standard) topology on $\overline \R$''' is the order topology $\tau$ associated to the ordering on $\overline \R$.
Definition:External Direct Product/General Definition
Let $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$ be algebraic structures. Let $\displaystyle \mathcal S_n = \prod_{k \mathop = 1}^n S_k$ be the cartesian product of $S_1, S_2, \ldots, S_n$. Let $\circledcirc_n$ be the operation induced on $\mathcal S_n$ by $\circ_1, \ldots, \circ_n$ defined as: :$\left({s_1, s_2, \ldots, s_n}\right) \circledcirc_n \left({t_1, t_2, \ldots, t_n}\right) := \begin{cases} s_1 \circ_1 t_1 & : n = 1 \\ \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2}\right) & : n = 2 \\ \left({\left({s_1, s_2, \ldots, s_{n-1} }\right) \circledcirc_{n-1} \left({t_1, t_2, \ldots, t_{n-1}}\right), s_n \circ_n t_n}\right) & : n > 2 \end{cases}$ for all ordered $n$-tuples in $\mathcal S_n$. That is: :$\left({s_1, s_2, \ldots, s_n}\right) \circledcirc_n \left({t_1, t_2, \ldots, t_n}\right) := \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$ The algebraic structure $\left({\mathcal S_n, \circledcirc_n}\right)$  is called the '''(external) direct product''' of $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$.
Definition:External Direct Product/Structures with Two Operations
Let $\left({S_1, +_1 ,\circ_1}\right), \left({S_2, +_2 ,\circ_2}\right), \ldots, \left({S_n, +_n ,\circ_n}\right)$ be algebraic structures with two operations. Let $\displaystyle S = \prod_{k \mathop = 1}^n S_k$ be as defined in cartesian product. The operation $+$ induced on $S$ by $+_1, \ldots, +_n$ is defined as: :$\left({s_1, s_2, \ldots, s_n}\right) + \left({t_1, t_2, \ldots, t_n}\right) = \left({s_1 +_1 t_1, s_2 +_2 t_2, \ldots, s_n +_n t_n}\right)$ The operation $\circ $ induced on $S$ by $\circ_1, \ldots, \circ_n$ is defined as: :$\left({s_1, s_2, \ldots, s_n}\right) \circ \left({t_1, t_2, \ldots, t_n}\right) = \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$ for all ordered $n$-tuples in $S$. The algebraic structure $\left({S, +, \circ}\right)$  is called the '''(external) direct product''' of $\left({S_1, +_1 ,\circ_1}\right), \left({S_2, +_2 ,\circ_2}\right), \ldots, \left({S_n, +_n ,\circ_n}\right)$.
Definition:Cartesian Product/Finite
Let $\sequence {S_n}$ be a sequence of sets.  The '''cartesian product''' of $\sequence {S_n}$ is defined as: :$\displaystyle \prod_{k \mathop = 1}^n S_k = \set {\tuple {x_1, x_2, \ldots, x_n}: \forall k \in \N^*_n: x_k \in S_k}$ It is also denoted $S_1 \times S_2 \times \cdots \times S_n$. Thus $S_1 \times S_2 \times \cdots \times S_n$ is the set of all ordered $n$-tuples $\tuple {x_1, x_2, \ldots, x_n}$ with $x_k \in S_k$.
Definition:Canonical Injection (Abstract Algebra)/General Definition
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''' $\displaystyle \inj_j: \struct {S_j, \circ_j} \to \prod_{i \mathop = 1}^n \struct {S_i, \circ_i}$ is defined as: :$\map {\inj_j} x = \tuple {e_1, e_2, \dotsc, e_{j - 1}, x, e_{j + 1}, \dotsc, e_n}$
Definition:Sturm-Liouville Theory
'''Sturm-Liouville theory''' is the branch of mathematical physics concerned with the eigenvalues and eigenfunctions arising from the Sturm-Liouville equation: :$\map {\dfrac \d {\d x} } {\map p x \dfrac {\d y} {\d x} } + \map q x y = -\lambda \map w x y$
Definition:Sturm-Liouville Equation
A classical '''Sturm-Liouville equation''' is a real second order ordinary linear differential equation of the form: :$ (1): \quad \displaystyle - \frac {\mathrm d} {\mathrm d x} \left({p \left({x}\right) \frac {\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 $b$. === Weight Function === The function $w \left({x}\right)$, which is sometimes called $r \left({x}\right)$, is called the '''weight function''' or '''density function'''.  === Eigenvalues === The value of $\lambda$ is not specified in the equation. Finding the values of $\lambda$ for which there exists a non-trivial solution of $(1)$ satisfying the boundary conditions is part of the problem called the Sturm-Liouville problem (S-L).  Such values of $\lambda$ when they exist are called the '''eigenvalues''' of the boundary value problem defined by $(1)$ and the prescribed set of boundary conditions. The corresponding solutions (for such a $\lambda$) are the '''eigenfunctions''' of this problem.
Definition:Column Space
Let $R$ be a ring. Let: :$\mathbf A_{m \times n} = \begin{bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \\ \end{bmatrix}$ be a matrix over $R$ such that every column is defined as a vector: :$\forall i: 1 \le i \le m: \begin {bmatrix} a_{1 i} \\ a_{2 i} \\ \vdots \\ a_{m i} \end {bmatrix} \in \mathbf V$ where $\mathbf V$ is some vector space. Then the '''column space of $\mathbf A$''' is the linear span of all such column vectors: :$\map {\mathrm C} {\mathbf A} = \map {\mathrm {span} } {\begin {bmatrix} a_{1 1} \\ a_{2 1} \\ \vdots \\ a_{m 1} \end {bmatrix}, \begin {bmatrix} a_{1 2} \\ a_{2 2} \\ \vdots \\ a_{m 2} \end {bmatrix}, \cdots, \begin {bmatrix} a_{1 n} \\ a_{2 n} \\ \vdots \\ a_{ mn} \end {bmatrix} }$
Definition:Row Space
Let $R$ be a ring. Let $\mathbf A$ be a matrix over $R$. Let $\mathbf A^\intercal$ be the transpose of $\mathbf A$.   Let the columns of $\mathbf A^\intercal$ be members of a vector space. The '''row space of $\mathbf A$''' is defined as the column space of $\mathbf A^\intercal$.
Definition:Simultaneous Linear Equations/Matrix Representation
A '''system of simultaneous linear equations''' can be expressed as: :$\mathbf A \mathbf x = \mathbf b$ where: :$\mathbf A = \begin {bmatrix} \alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} \\ \end {bmatrix}$,  $\mathbf x = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf b = \begin {bmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end {bmatrix}$ are matrices.
Definition:Simultaneous Equations/Solution Set
Consider the system of $m$ simultaneous equations in $n$ variables: :$\mathbb S := \forall i \in \set {1, 2, \ldots, m} : \map {f_i} {x_1, x_2, \ldots x_n} = \beta_i$ Let $\mathbb X$ be the set of ordered $n$-tuples: :$\set {\sequence {x_j}_{j \mathop \in \set {1, 2, \ldots, n} }: \forall i \in \set {1, 2, \ldots, m}: \map {f_i} {\sequence {x_j} } = \beta_i}$ which satisfies each of the equations in $\mathbb S$. Then $\mathbb X$ is called the '''solution set''' of $\mathbb S$.
Definition:Order Completion
Let $\struct {S, \preceq_S}$ be an ordered set. An ordered set $\struct {T, \preceq_T}$ is an '''order completion of $S$''' : :$(1):\quad S \subseteq T$ :$(2):\quad {\preceq_T \restriction_S} = {\preceq_S}$, where $\restriction$ denotes restriction :$(3):\quad \struct {T, \preceq_T}$ is a complete ordered set :$(4):\quad$ For all ordered sets $\struct {T', \preceq_{T'} }$ satisfying $(1), (2)$ and $(3)$, there is a unique order-preserving injection $\phi: T' \to T$
Definition:Complete Lattice/Definition 2
Let $\struct {S, \preceq}$ be an ordered set. Then $\struct {S, \preceq}$ is a '''complete lattice''' : :$\forall S' \subseteq S: \inf S', \sup S' \in S$ That is,  all subsets of $S$ have both a supremum and an infimum.
Definition:Left Null Space
Let $R$ be a ring. Let $\mathbf A$ be a matrix in the matrix space $\mathcal M_{m, n}\left({R}\right)$. Let $\mathbf A^\intercal$ be the transpose of $\mathbf A$. The '''left null space''' '''$\mathbf A$''' is defined as the null space of $\mathbf A^\intercal$.
Definition:Extended Real Addition
Let $\overline \R$ denote the extended real numbers. Define '''extended real addition''' or '''addition on $\overline \R$''', denoted $+_{\overline \R}: \overline \R \times \overline \R \to \overline \R$, by: :$\forall x,y \in \R: x +_{\overline \R} y := x +_\R y$ where $+_\R$ denotes real addition :$\forall x \in \R: x +_{\overline \R} \left({+\infty}\right) = \left({+\infty}\right) +_{\overline \R} x := +\infty$ :$\forall x \in \R: x +_{\overline \R} \left({-\infty}\right) = \left({-\infty}\right) +_{\overline \R} x := -\infty$ :$\left({+\infty}\right) +_{\overline \R} \left({+\infty}\right) := +\infty$ :$\left({-\infty}\right) +_{\overline \R} \left({-\infty}\right) := -\infty$ In particular, the expressions: :$\left({+\infty}\right) +_{\overline \R} \left({-\infty}\right)$ :$\left({-\infty}\right) +_{\overline \R} \left({+\infty}\right)$ are considered '''void''' and should be avoided. When no danger of confusion arises, $+_{\overline \R}$ is usually replaced with the more familiar $+$. From the definition of $+_{\overline \R}$ on bona fide real numbers, the name '''extended real addition''' is appropriate: the real addition is indeed extended.
Definition:Extended Real Subtraction
Let $\overline \R$ denote the extended real numbers. Define '''extended real subtraction''' or '''subtraction on $\overline \R$''', denoted $-_{\overline \R}: \overline \R \times \overline \R \to \overline \R$, by: :$\forall x, y \in \R: x -_{\overline \R} y := x -_{\R} y$ where $-_\R$ denotes real subtraction :$\forall x \in \R: x -_{\overline \R} \paren {+\infty} = \paren {-\infty} -_{\overline \R} x := -\infty$ :$\forall x \in \R: x -_{\overline \R} \paren {-\infty} = \paren {+\infty} -_{\overline \R} x := +\infty$ :$\paren {-\infty} -_{\overline \R} \paren {+\infty} := -\infty$ :$\paren {+\infty} -_{\overline \R} \paren {-\infty} := +\infty$ In particular, the expressions: :$\paren {+\infty} -_{\overline \R} \paren {+\infty}$ :$\paren {-\infty} -_{\overline \R} \paren {-\infty}$ are considered '''void''' and should be avoided.
Definition:Extended Real Multiplication
Let $\overline \R$ denote the extended real numbers. Define '''extended real multiplication''' or '''multiplication on $\overline \R$''', denoted $\times_{\overline \R}: \overline \R \times \overline \R \to \overline \R$, by: :$\forall x \in \overline \R: x \times_{\overline \R} 0 = 0 \times_{\overline \R} x = 0$ :$\forall x, y \in \R: x \times_{\overline \R} y := x \times_\R y$ where $\times_\R$ denotes real multiplication :$\forall x \in \R_{>0}: x \times_{\overline \R} \paren {+\infty} = \paren {+\infty} \times_{\overline \R} x := +\infty$ :$\forall x \in \R_{<0}: x \times_{\overline \R} \paren {+\infty} = \paren {+\infty} \times_{\overline \R} x := -\infty$ :$\forall x \in \R_{>0}: x \times_{\overline \R} \paren {-\infty} = \paren {-\infty} \times_{\overline \R} x := -\infty$ :$\forall x \in \R_{<0}: x \times_{\overline \R} \paren {-\infty} = \paren {-\infty} \times_{\overline \R} x := +\infty$ :$\paren {+\infty} \times_{\overline \R} \paren {+\infty} := +\infty$ :$\paren {-\infty} \times_{\overline \R} \paren {-\infty} := +\infty$ :$\paren {+\infty} \times_{\overline \R} \paren {-\infty} := -\infty$ :$\paren {-\infty} \times_{\overline \R} \paren {+\infty} := -\infty$ When no danger of confusion arises, $\times_{\overline \R}$ is usually replaced with the more familiar $\times$, or even suppressed. From the definition of $\times_{\overline \R}$ on bona fide real numbers, the name '''extended real multiplication''' is appropriate: the real multiplication is indeed extended.
Definition:Internal Direct Product/General Definition
Let $\left({S_1, \circ {\restriction_{S_1}}}\right), \ldots, \left({S_n, \circ {\restriction_{S_n}}}\right)$ be closed algebraic substructures of an algebraic structure $\left({S, \circ}\right)$ where $\circ {\restriction_{S_1}}, \ldots, \circ {\restriction_{S_n}}$ are the operations induced by the restrictions of $\circ$ to $S_1, \ldots, S_n$ respectively. The structure $\left({S, \circ}\right)$ is the '''internal direct product of $\left \langle {S_n} \right \rangle$''' if the mapping: :$\displaystyle C: \prod_{k \mathop = 1}^n S_k \to S: C \left({s_1, \ldots, s_n}\right) = \prod_{k \mathop = 1}^n s_k$ is an isomorphism from the cartesian product $\left({S_1, \circ {\restriction_{S_1}}}\right) \times \cdots \times \left({S_n, \circ {\restriction_{S_n}}}\right)$ onto $\left({S, \circ}\right)$. The operation $\circ$ on $S$ is the operation induced on $S$ by $\circ {\restriction_{S_1}}, \circ {\restriction_{S_2}}, \ldots, \circ {\restriction_{S_n}}$.
Definition:Measurable Mapping
Let $\left({X, \Sigma}\right)$ and $\left({X', \Sigma'}\right)$ be measurable spaces. A mapping $f: X \to X'$ is said to be '''$\Sigma \, / \, \Sigma'$-measurable''' iff: :$\forall E' \in \Sigma': f^{-1} \left({E'}\right) \in \Sigma$ That is, iff the preimage of every measurable set under $f$ is again measurable.
Definition:Operation Induced by Direct Product
Let $\left({S_1, \circ_1}\right)$ and $\left({S_2, \circ_2}\right)$ be algebraic structures. Let $S = S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$. Then the '''operation induced on $S$ by $\circ_1$ and $\circ_2$''' is the operation $\circ$ defined as: :$\left({s_1, s_2}\right) \circ \left({t_1, t_2}\right) := \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2}\right)$ for all ordered pairs in $S$. === General Definition === {{:Definition:Operation Induced by Direct Product/General Definition}}
Definition:Translation Mapping
=== In an Abelian Group === {{:Definition:Translation Mapping/Abelian Group}} === In a Euclidean Space === {{:Definition:Translation Mapping/Euclidean Space}} === In an Affine Space === {{:Definition:Translation Mapping/Affine Space}}
Definition:Operation Induced by Direct Product/General Definition
Let $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$ be algebraic structures. Let $\displaystyle \mathcal S_n = \prod_{k \mathop = 1}^n S_k$ be the cartesian product of $S_1, S_2, \ldots, S_n$. Then the '''operation induced on $\mathcal S_n$ by $\circ_1, \ldots, \circ_n$''' is the operation $\circledcirc_n$ defined as: :$\left({s_1, s_2, \ldots, s_n}\right) \circledcirc_n \left({t_1, t_2, \ldots, t_n}\right) := \begin{cases} s_1 \circ_1 t_1 & : n = 1 \\ \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2}\right) & : n = 2 \\ \left({\left({s_1, s_2, \ldots, s_{n-1} }\right) \circledcirc_{n-1} \left({t_1, t_2, \ldots, t_{n-1}}\right), s_n \circ_n t_n}\right) & : n > 2 \end{cases}$ for all ordered $n$-tuples in $\mathcal S_n$. That is: :$\left({s_1, s_2, \ldots, s_n}\right) \circledcirc_n \left({t_1, t_2, \ldots, t_n}\right) := \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$
Definition:Pre-Image Sigma-Algebra
=== On Domain === {{:Definition:Pre-Image Sigma-Algebra/Domain}} === On Codomain === {{:Definition:Pre-Image Sigma-Algebra/Codomain}}
Definition:Pre-Image Sigma-Algebra/Codomain
Let $X, X'$ be sets, and let $f: X \to X'$ be a mapping. Let $\Sigma$ be a $\sigma$-algebra on $X$. Then the '''pre-image $\sigma$-algebra (of $\Sigma$) on the codomain of $f$''' is defined as: :$\Sigma' := \left\{{E' \subseteq X': f^{-1} \left({E'}\right) \in \Sigma}\right\}$
Definition:Pointwise Operation
Let $S$ be a set. Let $\struct {T, \circ}$ be an algebraic structure. Let $T^S$ be the set of all mappings from $S$ to $T$. Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings. Then the operation $f \oplus g$ is defined on $T^S$ as follows: :$f \oplus g: S \to T: \forall x \in S: \map {\paren {f \oplus g} } x = \map f x \circ \map g x$ The operation $\oplus$ is called the '''pointwise operation on $T^S$ induced by $\circ$'''. === Induced Structure === {{:Definition:Pointwise Operation/Induced Structure}}