hcju/proofwikips huggingface dataset:
Definition: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 the '''monotone class generated by $\mathcal G$''', $\mathfrak m \left({\mathcal G}\right)$, is the smallest monotone class on $X$ that contains $\mathcal G$. That is, $\mathfrak m \left({\mathcal G}\right)$ is subject to: :$(1):\quad \mathcal G \subseteq \mathfrak m \left({\mathcal G}\right)$ :$(2):\quad \mathcal G \subseteq \mathcal M \implies \mathfrak m \left({\mathcal G}\right) \subseteq \mathcal M$ for any monotone class $\mathcal M$ on $X$ In fact, $\mathfrak m \left({\mathcal G}\right)$ always exists, and is unique, as proved on Existence and Uniqueness of Monotone Class Generated by Collection of Subsets. === Generator === One says that $\mathcal G$ is a '''generator''' for $\mathfrak m \left({\mathcal G}\right)$.
Definition:Topology Induced by Metric
Let $M = \left({A, d}\right)$ be a metric space. === Definition 1 === {{:Definition:Topology Induced by Metric/Definition 1}} === Definition 2 === {{:Definition:Topology Induced by Metric/Definition 2}}
Definition:Pre-Measure
Let $X$ be a set. Let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Let $\varnothing \in \mathcal S$. Let $\mu: \mathcal S \to \overline{\R}_{\ge 0}$ be a mapping, where $\overline{\R}_{\ge 0}$ denotes the set of positive extended real numbers. Then $\mu$ is said to be a '''pre-measure''' iff it satisfies the following conditions: :$(1):\quad \mu \left({\varnothing}\right) = 0$ :$(2):\quad$ For every sequence $\left({A_n}\right)_{n \in \N}$ of pairwise disjoint sets in $\mathcal S$ with $\displaystyle \bigcup_{n \mathop \in \N} A_n \in \mathcal S$: ::$\displaystyle \mu \left({\bigcup_{n \mathop \in \N} A_n}\right) = \sum_{n \mathop \in \N} \mu \left({A_n}\right)$ :that is, that $\mu$ is countably additive.
Definition:Finite Measure
Let $\mu$ be a measure on a measurable space $\struct {X, \Sigma}$. Then $\mu$ is said to be a '''finite measure''' : :$\map \mu X < \infty$
Definition:Finite Measure Space
A measure space $\struct {X, \Sigma, \mu}$ is said to be '''finite'''  $\mu$ is a finite measure.
Definition:Exhausting Sequence of Sets
Let $\left\langle{S_k}\right \rangle_{k \in \N}$ be a nested sequence of subsets of $S$ such that: :$(1):\quad \forall k \in \N: S_k \subseteq S_{k + 1}$ :$(2):\quad \displaystyle \bigcup_{k \mathop \in \N} S_k = S$ Then $\left\langle{S_k}\right \rangle_{k \in \N}$ is an '''exhausting sequence of sets (in $\mathcal S$)'''. That is, it is an increasing sequence of subsets of $S$, whose union is $S$. It is common to write $\left\langle{S_k}\right\rangle_{k \in \N} \uparrow S$ to indicate an '''exhausting sequence of sets'''. Here, the $\uparrow$ denotes a limit of an increasing sequence.
Definition:Increasing Sequence of Sets
Let $\sequence {S_k}_{k \mathop \in \N}$ be a nested sequence of subsets of $S$ such that: :$\forall k \in \N: S_k \subseteq S_{k + 1}$ Then $\sequence {S_k}_{k \mathop \in \N}$ is an '''increasing sequence of sets (in $\SS$)'''.
Definition:Sigma-Finite Measure
Let $\mu$ be a measure on a measurable space $\left({X, \Sigma}\right)$. Then $\mu$ is said to be a '''$\sigma$-finite''' (or '''sigma-finite''') '''measure''' iff there exists an exhausting sequence $\left({E_n}\right)_{n \in \N}$ in $\Sigma$, subject to: :$\forall n \in \N: \mu \left({E_n}\right) < \infty$
Definition:Sigma-Finite Measure Space
A measure space $\left({X, \Sigma, \mu}\right)$ is said to be '''$\sigma$-finite''' (or '''sigma-finite''') iff $\mu$ is a $\sigma$-finite measure.
Definition:G-Submodule
Let $\left({G, \cdot}\right)$ be a finite group. Let $\left({V, \phi}\right)$ be a $G$-module. Let $W$ be a vector subspace of $V$. If $\phi$ is a linear group action when restricted to $G \times W \subseteq G \times V$. Then $(W,\phi_W)$; where $\phi_W$ is the restriction of $\phi$ on $G\times W$, is called a $G$-submodule of $(V,\phi)$.
Definition:G-Module Homomorphism
Let $\left({G, \cdot}\right)$ be a group. Let $\left({V, \phi}\right)$ and $\left({W, \mu}\right)$ be $G$-modules. Then a linear mapping $f: V \to W$ is called a '''$G$-module homomorphism''' : :$\forall g \in G: \forall v \in V: f \left({\phi \left({g, v}\right)}\right) = \mu \left({g, f \left({v}\right)}\right)$
Definition:Null Space
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}$,  $\mathbf x_{n \times 1} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf 0_{m \times 1} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$ be matrices where each column is a member of a real vector space. The set of all solutions to $\mathbf A \mathbf x = \mathbf 0$: :$\map {\operatorname N} {\mathbf A} = \set {\mathbf x \in \R^n : \mathbf {A x} = \mathbf 0}$ is called the '''null space''' of $\mathbf A$.
Definition:G-Module
Let $\left({V, +, \cdot}\right)$ be a vector space over a field $\left({k, \oplus, \circ}\right)$. Let $G$ be a group. Let $\phi : G \times V \to V$ be an linear group action of $G$ on $V$. Then $(V,\phi)$ is called a $G$-module.
Definition:Limit of Increasing Sequence of Sets
Let $\left({S_n}\right)_{n \in \N}$ be an increasing sequence of sets. Let $S = \displaystyle \bigcup_{n \mathop \in \N} S_n$. Then $S$ is said to be the '''limit''' of $\left({S_n}\right)_{n \in \N}$, and one writes $S_n \uparrow S$.
Definition:Decreasing Sequence of Sets
Let $\sequence {S_k}_{k \mathop \in \N}$ be a nested sequence of subsets of $S$ such that: :$\forall k \in \N: S_k \supseteq S_{k + 1}$ Then $\sequence {S_k}_{k \mathop \in \N}$ is a '''decreasing sequence of sets (in $\SS$)'''.
Definition:Limit of Decreasing Sequence of Sets
Let $\left({S_n}\right)_{n \in \N}$ be a decreasing sequence of sets. Let $S = \displaystyle \bigcap_{n \mathop \in \N} S_n$. Then $S$ is said to be the '''limit''' of $\left({S_n}\right)_{n \in \N}$, and one writes $S_n \downarrow S$.
Definition:Reducible G-Module
Let $M$ be a $G$-module. Then $M$ is '''reducible'''  the corresponding linear representation is reducible.
Definition:Edge Contraction
Let $G$ be an undirected graph. Let $e \in \map E G$ be an edge of $G$. Then the '''graph obtained by contracting $e$ in $G$''', denoted by $G / e$, is the graph $H$ defined by: :$\map V H = \paren {\map V G \setminus e} \cup \set v$ :$\map E H = \set {f \in \map E G: f \cap e = \O} \cup \set {u v: \exists f \in \map E G: u \in f \setminus e, f \cap e \ne \O}$ where $v \notin \map V G$ is a fresh object. {{explain|A lot of the above - see talk page (in prep)}} Informally, it is the graph obtained from $G$ by replacing the vertices incident to $e$ with a single vertex adjacent to all their neighbors.
Definition:Dirac Measure
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $x \in X$ be any point in $X$. Then the '''Dirac measure at $x$''', denoted $\delta_x$ is the measure defined by: :$\delta_x: \Sigma \to \overline \R, \ \delta_x \left({E}\right) := \begin{cases}0 & \text{if } x \notin E \\ 1 & \text{if } x \in E\end{cases}$ where $\overline \R$ denotes the extended set of real numbers. That $\delta_x$ actually is a measure is shown on Dirac Measure is Measure. In fact, Dirac measure is a probability measure.
Definition:Enumeration
=== Finite Sets === {{:Definition:Enumeration/Finite}} === Countably Infinite Sets === {{:Definition:Enumeration/Countably Infinite}}
Definition:Discrete Probability Measure
Let $\Omega$ be a countable set. Let $\mathcal P \left({\Omega}\right)$ be its power set, regarded as a $\sigma$-algebra. Let $\left({p_\omega}\right)_{\omega \in \Omega} \subseteq \left[{0 \,.\,.\, 1}\right]$ be a subset of the closed unit interval in $\R$, indexed by $\Omega$. Suppose that $\displaystyle \sum_{\omega \mathop \in \Omega} p_\omega = 1$. {{explain|link to definition of sum over countable set}} The '''discrete probability measure on $\Omega$''', denoted $P$, is the mapping defined by: :$\displaystyle P: \mathcal P \left({\Omega}\right) \to \overline \R, \ P \left({S}\right) = \sum_{\omega \mathop \in \Omega} p_\omega \delta_\omega \left({S}\right)$ where $\overline \R$ denotes the extended real numbers, and $\delta_\omega$ is the Dirac measure at $\omega$. From this definition, it is seen that the name '''discrete probability measure''' is compatible with the notion of discrete measure, as $\Omega$ is countable. {{MissingLinks|to eg. Definition:Discrete Sample Space and connected stuff}} === Discrete Probability Space === The measure space $\left({\Omega, \mathcal P \left({\Omega}\right), P}\right)$ is called '''discrete probability space'''.
Definition:Proper G-Submodule
Let $\left({V, \phi}\right)$ be a $G$-module. A $G$-submodule of $V$ is called '''proper''' iff it is a proper vector subspace of $V$.  hu0luq718d6r29qzya71z71pi3o2iro
Definition:Trivial G-Module
Let $\left({V, \phi}\right)$ be a $G$-module. Then $\left({V, \phi}\right)$ is said to be '''trivial''' iff $V = \left\{{\mathbf 0}\right\}$.  bhlwwt3n0a6zqpyzdyhlznjnsw4gdk3
Definition:Irreducible (Representation Theory)/G-Module
A $G$-module is '''irreducible'''  the corresponding linear representation is irreducible.
Definition:Equivalent Linear Representations
Let $ \struct {G, \cdot}$ be a group. Consider two linear representations $\rho: G \to \GL V$ and $\rho': G \to \GL W$ of $G$. Then $\rho$ and $\rho'$ are called '''equivalent (linear representations)'''  their correspondent $G$-modules using Correspondence between Linear Group Actions and Linear Representations are isomorphic. {{rewrite|above line needs rewriting, but I can't come up with a suitable replacement}} {{explain|The "isomorphic" link goes to a generic "abstract algebra" page, but I believe no actual definition has been made for an isomorphism between two G-modules.}}
Definition:Infinite Measure
Let $\left({X, \Sigma}\right)$ be a measurable space. Then the '''infinite measure''' is the measure defined by: :$\mu: \Sigma \to \overline \R, \ \mu \left({E}\right) := \begin{cases} 0 & : \text{if $E = \varnothing$} \\ +\infty & : \text{otherwise}\end{cases}$ where $\overline \R$ denotes the extended real numbers.
Definition:Null Measure
Let $\struct {X, \Sigma}$ be a measurable space. Then the '''null measure''' is the measure defined by: :$\mu: \Sigma \to \overline \R: \map \mu E := 0$ where $\overline \R$ denotes the extended real numbers.
Definition:Preimage/Relation
Let $\RR \subseteq S \times T$ be a relation. Let $\RR^{-1} \subseteq T \times S$ be the inverse relation to $\RR$, defined as: :$\RR^{-1} = \set {\tuple {t, s}: \tuple {s, t} \in \RR}$ === Preimage of Element === {{:Definition:Preimage/Relation/Element}} === Preimage of Subset === {{:Definition:Preimage/Relation/Subset}} === Preimage of Relation === {{:Definition:Preimage/Relation/Relation}}
Definition:Preimage/Mapping
Let $f: S \to T$ be a mapping. Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, considered as a relation: :$f^{-1} = \set {\tuple {t, s}: \map f s = t}$ === Preimage of Element === {{:Definition:Preimage/Mapping/Element}} === Preimage of Subset === {{:Definition:Preimage/Mapping/Subset}} === Preimage of Mapping === {{:Definition:Preimage/Mapping/Mapping}}
Definition:Preimage/Relation/Element
Every $s \in S$ such that $\tuple {s, t} \in \RR$ is called '''a preimage of $t$'''. In some contexts, it is not individual elements that are important, but ''all'' elements of $S$ which are of interest. Thus '''the preimage of $t \in T$''' is defined as: :$\map {\RR^{-1} } t := \set {s \in S: \tuple {s, t} \in \RR}$ This can also be written: :$\map {\RR^{-1} } t := \set {s \in \Img {\RR^{-1} }: \tuple {t, s} \in \RR^{-1} }$ That is, '''the preimage of $t$ under $\RR$''' is the image of $t$ under $\RR^{-1}$.
Definition:Preimage/Relation/Subset
Let $Y \subseteq T$. The '''preimage of $Y$ under $\RR$''' is defined as: :$\RR^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \tuple {s, t} \in \RR}$ That is, the '''preimage of $Y$ under $\RR$''' is the image of $Y$ under $\RR^{-1}$: :$\RR^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \tuple {t, s} \in \RR^{-1} }$ If no element of $Y$ has a '''preimage''', then $\RR^{-1} \sqbrk Y = \O$.
Definition:Preimage/Relation/Relation
The '''preimage''' of $\RR \subseteq S \times T$ is: :$\Preimg \RR := \RR^{-1} \sqbrk T = \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$
Definition:Preimage/Mapping/Element
Every $s \in S$ such that $\map f s = t$ is called '''a preimage''' of $t$. '''The preimage''' of an element $t \in T$ is defined as: :$\map {f^{-1} } t := \set {s \in S: \map f s = t}$ This can also be expressed as: :$\map {f^{-1} } t := \set {s \in \Img {f^{-1} }: \tuple {t, s} \in f^{-1} }$ That is, the '''preimage of $t$ under $f$''' is the image of $t$ under $f^{-1}$.
Definition:Preimage/Mapping/Subset
Let $Y \subseteq T$.
Definition:Preimage/Mapping/Mapping
The '''preimage of $f$''' is defined as: :$\Preimg f := \set {s \in S: \exists t \in T: f \paren s = t}$ That is: :$\Preimg f := f^{-1} \sqbrk T$ where $f^{-1} \sqbrk T$ is the image of $T$ under $f^{-1}$. In this context, $f^{-1} \subseteq T \times S$ is the the inverse of $f$. It is a relation but not necessarily itself a mapping.
Definition:Image (Set Theory)/Relation/Relation
The '''image''' of $\RR$ is the set: :$\Img \RR := \RR \sqbrk S = \set {t \in T: \exists s \in S: \tuple {s, t} \in \RR}$
Definition:Image (Set Theory)/Mapping/Mapping
Let $f: S \to T$ be a mapping. <section notitle="true" name="tc"> === Definition 1 === {{:Definition:Image (Set Theory)/Mapping/Mapping/Definition 1}} === Definition 2 === {{:Definition:Image (Set Theory)/Mapping/Mapping/Definition 2}} </section>
Definition:Arbitrarily Large
Let $P$ be a property of real numbers. We say that '''$P \left({x}\right)$ holds for arbitrarily large $x$''' (or '''there exist arbitrarily large $x$ such that $P \left({x}\right)$ holds''') : :$\forall a \in \R: \exists x \in \R: x \ge a: P \left({x}\right)$ That is: :''For any real number $a$, there exists a (real) number not less than $a$ such that the property $P$ holds.'' or, more informally and intuitively: :''However large a number you can think of, there will be an even larger one for which $P$ still holds.''
Definition:Valuation
Let $\left({R, +, \cdot}\right)$ be a ring. A '''valuation on $R$''' is a mapping: : $\nu: R \to \Z \cup \left\{{+\infty}\right\}$ which fulfils the '''valuation axioms''': {{:Definition:Valuation Axioms}}
Definition:P-adic Valuation
Let $p \in \N$ be a prime number. === Integers === {{:Definition:P-adic Valuation/Integers}} === Rational Numbers === {{:Definition:P-adic Valuation/Rational Numbers}}
Definition:Angle Between Vectors
Let $\mathbf v, \mathbf w$ be two non-zero vectors in $\R^n$. === Case 1 === Suppose that $\mathbf v$ and $\mathbf w$ are not scalar multiples of each other: :$\neg \exists \lambda \in \R: \mathbf v = \lambda \mathbf w$ Then the '''angle between $\mathbf v$ and $\mathbf w$''' is defined as follows: Describe a triangle with lengths corresponding to: :$\norm {\mathbf v}, \norm {\mathbf w}, \norm {\mathbf v - \mathbf w}$ where $\norm {\, \cdot \,}$ denotes vector length: :600px The angle formed between the two sides with lengths $\norm {\mathbf v}$ and $\norm {\mathbf w}$ is called the '''angle between vectors $\mathbf v$ and $\mathbf w$'''. By convention, the angle is taken between $0$ and $\pi$. === Case 2 === Suppose that $\mathbf v$ and $\mathbf w$ ''are'' scalar multiples of each other: :$\exists \lambda \in \R: \mathbf v = \lambda \mathbf w$ As $\mathbf v$ and $\mathbf w$ as non-zero, $\lambda \ne 0$. If $\lambda > 0$, then the angle between $\mathbf v$ and $\mathbf w$ is defined as a zero angle, that is: :$\theta = 0$ If $\lambda < 0$, then the angle between $\mathbf v$ and $\mathbf w$ is defined as a straight angle, that is: :$\theta = \pi$
Definition:P-adic Number
Let $p$ be any prime number. Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$. By P-adic Norm is Non-Archimedean Norm then $\norm {\,\cdot\,}_p$ is a non-archimedean norm on $\Q$ and the pair $\struct {\Q, \norm {\,\cdot\,}_p}$ is a valued field. === $p$-adic Norm Completion of Rational Numbers === {{:Definition:P-adic Number/P-adic Norm Completion of Rational Numbers}} === Quotient of Cauchy Sequences in $p$-adic Norm === {{:Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm}} === $p$-adic Norm on $\Q_p$ === {{:Definition:P-adic Norm on P-adic Numbers}}
Definition:Darboux Integral/Geometric Interpretation
The expression $\displaystyle \int_a^b \map f x \rd x$ can be (and frequently is) interpreted as '''the area under the graph'''.  This follows from the definition of the definite integral as a sum of the product of the lengths of intervals and the "height" of the function being integrated in that interval and the formula for the area of a rectangle. A depiction of the lower and upper sums illustrates this: :350px  350px It can intuitively be seen that as the number of points in the subdivision increases, the more "accurate" the lower and upper sums become. Also note that if the graph is below the $x$-axis, the signed area under the graph becomes negative.
Definition:Intersection Measure
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $F \in \Sigma$. Then the '''intersection measure (of $\mu$ by $F$)''' is the mapping $\mu_F: \Sigma \to \overline{\R}$, defined by: :$\mu_F \left({E}\right) = \mu \left({E \cap F}\right)$ It is in fact a measure on $\left({X, \Sigma}\right)$, as shown on Intersection Measure is Measure. {{expand|verify name, Schilling gives none}}
Definition:Perpendicular (Linear Algebra)
Let $\mathbf u$, $\mathbf v$ be non-zero vectors in the Euclidean space $\R^n$. :$\mathbf u$ and $\mathbf v$ are '''perpendicular'''  the angle between them is a right angle.
Definition:Complete Measure Space
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Suppose that the family of $\mu$-null sets $\mathcal{N}_{\mu}$ satisfies the following condition: :$\forall N \in \mathcal{N}_{\mu}: \forall M \subseteq N: M \in \mathcal{N}_{\mu}$ That is, any subset of a $\mu$-null set is again a $\mu$-null set. Then $\left({X, \Sigma, \mu}\right)$ is said to be a '''complete measure space'''.
Definition:Completion (Measure Space)
Let $\left({X, \Sigma, \mu}\right), \left({\tilde X, \Sigma^*, \bar \mu}\right)$ be measure spaces. Then '''$\left({\tilde X, \Sigma^*, \bar \mu}\right)$ is a completion of $\left({X, \Sigma, \mu}\right)$''' or '''$\left({\tilde X, \Sigma^*, \bar \mu}\right)$ completes $\left({X, \Sigma, \mu}\right)$''' iff the following conditions hold: :$(1):\quad \left({\tilde X, \Sigma^*, \bar \mu}\right)$ is a complete measure space :$(2):\quad \tilde X = X$ :$(3):\quad \Sigma$ is a sub-$\sigma$-algebra of $\Sigma^*$ :$(4):\quad \forall E \in \Sigma: \bar \mu \left({E}\right) = \mu \left({E}\right)$, i.e. $\bar \mu \restriction_{\Sigma} = \mu$
Definition:Restricted Measure
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\Sigma'$ be a sub-$\sigma$-algebra of $\Sigma$. Then the '''restricted measure on $\Sigma'$''' or the '''restriction of $\mu$ to $\Sigma'$''' is the mapping $\nu: \Sigma \to \overline{\R}$ defined by: :$\forall E' \in \Sigma': \nu \left({E'}\right) = \mu \left({E'}\right)$ That is, $\nu$ is the restriction $\mu \restriction_{\Sigma'}$.
Definition:Dynkin System
Let $X$ be a set, and let $\mathcal D \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then $\mathcal D$ is called a '''Dynkin system (on $X$)'''  it satisfies the following conditions: :$(1):\quad X \in \mathcal D$ :$(2):\quad \forall D \in \mathcal D: X \setminus D \in \mathcal D$ :$(3):\quad$ For all pairwise disjoint sequences $\left({D_n}\right)_{n \in \N}$ in $\mathcal D$, $\displaystyle \bigcup_{n \mathop \in \N} D_n \in \mathcal D$
Definition: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 the '''Dynkin system generated by $\mathcal G$''', denoted $\delta \left({\mathcal G}\right)$, is the smallest Dynkin system on $X$ that contains $\mathcal G$. That is, $\delta \left({\mathcal G}\right)$ is subject to: :$(1):\quad \mathcal G \subseteq \delta \left({\mathcal G}\right)$ :$(2):\quad \mathcal G \subseteq \mathcal D \implies \delta \left({\mathcal G}\right) \subseteq \mathcal D$ for any Dynkin system $\mathcal D$ on $X$ In fact, $\delta \left({\mathcal G}\right)$ always exists, and is unique, as proved on Existence and Uniqueness of Dynkin System Generated by Collection of Subsets. === Generator === One says that $\mathcal G$ is a '''generator''' for $\delta \left({\mathcal G}\right)$.
Definition:Central Product
Let $G$ and $H$ be groups. Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively. Let: :$Z \cong W$ where $\cong$ denotes isomorphism. Let such a group isomorphism be $\theta: Z \to W$. Let $X$ be the set defined as: :$X = \set {\tuple {x, \map \theta x^{-1} }: x \in Z}$ Then the quotient group $\struct {G \times H} / X$ is denoted $\struct {G \times_\theta H}$ and is called the '''central product of $G$ and $H$ via $\theta$'''.
Definition:Coset Product
Let $\struct {G, \circ}$ be a group. Let $N$ be a normal subgroup of $G$. Let $a, b \in G$. The '''coset product''' of $a \circ N$ and $b \circ N$ is defined as: :$\paren {a \circ N} \circ \paren {b \circ  N} = \paren {a \circ b} \circ N$ where $a \circ N$ and $b \circ N$ are the left cosets of $a$ and $b$ by $N$.
Definition:Translation-Invariant Measure
Let $\mu$ be a measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$. Then $\mu$ is said to be '''translation-invariant''' or '''invariant under translations''' iff: :$\forall x \in \R^n, \forall B \in \mathcal B: \mu \left({x + B}\right) = \mu \left({B}\right)$ where $x + B$ is the set $\left\{{x + b: b \in B}\right\}$.  nid2a2b6pzp4gwqrs3jjup0l3wgnkzz
Definition:Invariant Measure
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\theta: X \to X$ be an $\Sigma / \Sigma$-measurable mapping. Then $\mu$ is said to be a '''$\theta$-invariant measure''' or to be '''invariant under $\theta$''' iff: :$\forall E \in \Sigma: \mu \left({\theta^{-1} \left({E}\right) }\right) = \mu \left({E}\right)$ In terms of a pushforward measure, this can be concisely formulated as: :$\theta_* \mu = \mu$
Definition:Independent Sigma-Algebras
Let $\struct {\Omega, \EE, \Pr}$ be a probability space. Let $\Sigma$ and $\Sigma'$ be sub-$\sigma$-algebras of $\EE$. Then $\Sigma$ and $\Sigma'$ are said to be '''($\Pr$-)independent''' : :$\forall E \in \Sigma, E' \in \Sigma': \map \Pr {E \cap E'} = \map \Pr E \map \Pr {E'}$
Definition:Stable under Intersection
Let $X$ be a set, and let $\SS \subseteq \powerset X$ be a collection of subsets of $X$. Then $\SS$ is said to be '''stable under intersection(s)''', or simply '''$\cap$-stable''', : :$\forall S, T \in \SS: S \cap T \in \SS$
Definition:Sub-Sigma-Algebra
Let $X$ be a set, and let $\mathcal A, \mathcal B$ be $\sigma$-algebras on $X$. Then $\mathcal B$ is said to be a '''sub-sigma-algebra''' or '''sub-$\sigma$-algebra''' of $\mathcal A$ iff $\mathcal B \subseteq \mathcal A$.  501i2ym6ibfjwwspmzl7ljcqmez1jpm
Definition:Lebesgue Pre-Measure
Let $\mathcal J_{ho}$ be the collection of half-open $n$-rectangles. '''$n$-dimensional Lebesgue pre-measure''' is the mapping $\lambda^n: \mathcal J_{ho} \to \overline \R_{\ge 0}$ given by: :$\displaystyle \lambda^n \left({ \left[[{\mathbf a \,.\,.\, \mathbf b}\right)) }\right) = \prod_{i \mathop = 1}^n \left({b_i - a_i}\right)$ where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.
Definition:Diffuse Measure
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. The measure $\mu$ is said to be '''diffuse''' or '''non-atomic'''  there are no $\mu$-atoms.