hcju/proofwikips huggingface dataset:
Definition:Sum of Cardinals
Let $A$ and $B$ be sets. Let $\mathbf a$ and $\mathbf b$ be the cardinals associated with $A$ and $B$ respectively. Then the '''sum of $\mathbf a$ and $\mathbf b$''' is defined as: :$\mathbf a + \mathbf b = \operatorname{Card} \left({A \sqcup B}\right)$ where: :$A \sqcup B$ denotes the disjoint union of $A$ and $B$ :$\operatorname{Card} \left({A \sqcup B}\right)$ denotes the cardinal associated with $A \cup B$.
Definition:Self-Inverse Element
Let $\struct {S, \circ}$ be a monoid whose identity element is $e$. Let $x \in S$ be an element of $S$. === Definition 1 === {{:Definition:Self-Inverse Element/Definition 1}} === Definition 2 === {{:Definition:Self-Inverse Element/Definition 2}}
Definition:Finite Cardinal
Let $\mathbf a$ be a cardinal. Then $\mathbf a$ is described as '''finite''' : :$\mathbf a < \mathbf a + \mathbf 1$ where $\mathbf 1$ is (cardinal) one. That is, such that $\mathbf a \ne \mathbf a + \mathbf 1$.
Definition:Infinite Cardinal
Let $\mathbf a$ be a cardinal. Then $\mathbf a$ is described as '''infinite''' : :$\mathbf a = \mathbf a + \mathbf 1$ where $\mathbf 1$ is (cardinal) one.
Definition:Convolution of Measurable Functions
Let $f, g: \R^n \to \R$ be $\mathcal B^n$-measurable functions such that for all $x \in \R^n$: :$\displaystyle \int_{\R^n} f \left({x - y}\right) g \left({y}\right) \, \mathrm d \lambda^n \left({y}\right)$ is finite. The '''convolution of $f$ and $g$''', denoted $f * g$, is the mapping defined by: :$\displaystyle f * g: \R^n \to \R, f * g \left({x}\right) := \int_{\R^n} f \left({x - y}\right) g \left({y}\right) \, \mathrm d \lambda^n \left({y}\right)$
Definition:Convolution of Measurable Function and Measure
Let $\mu$ be a measure on the Borel $\sigma$-algebra $\mathcal B^n$ on $\R^n$. Let $f: \R^n \to \R$ be a $\mathcal B^n$-measurable function such that for all $x \in \R^n$: :$\displaystyle \int_{\R^n} f \left({x - y}\right) \, \mathrm d \mu \left({y}\right)$ is finite. The '''convolution of $f$ and $\mu$''', denoted $f * \mu$, is the mapping defined by: :$\displaystyle f * \mu: \R^n \to \R, f * \mu \left({x}\right) := \int_{\R^n} f \left({x - y}\right) \, \mathrm d \mu \left({y}\right)$
Definition:Convolution of Measures
Let $\mu$ and $\nu$ be measures on the Borel $\sigma$-algebra $\mathcal B^n$ on $\R^n$. The '''convolution of $\mu$ and $\nu$''', denoted $\mu * \nu$, is the measure defined by: :$\displaystyle \mu * \nu: \mathcal B^n \to \overline \R, \mu * \nu \left({B}\right) := \int_{\R^n} \chi_B \left({x + y}\right) \, \mathrm d \mu \left({x}\right) \, \mathrm d \nu \left({y}\right)$ where $\chi_B$ is the characteristic function of $B$.
Definition:Group Direct Product/General Definition
Let $\family {\struct {G_i, \circ_i} }_{i \mathop \in I}$ be a family of groups. Let $\displaystyle G = \prod_{i \mathop \in I} G_i$ be their cartesian product. Let $\circ$ be the operation defined on $G$ as: :$\circ := \family {g_i}_{i \mathop \in I} \circ \family {h_i}_{i \mathop \in I} = \family {g_i \circ_i h_i}_{i \mathop \in I}$ for all sequences in $G$. The group $\struct {G, \circ}$  is called the '''(external) direct product''' of $\family {\struct {G_i, \circ_i} }_{i \mathop \in I}$.
Definition:Hölder Continuous
Let $\left({M_1, d_1}\right)$ and $\left({M_2, d_2}\right)$ be metric spaces. Let $\alpha \in \R_{\ge 0}$ be a positive real number. A mapping $f: M_1 \to M_2$ is said to be '''$\alpha$-Hölder continuous''' : :$\exists L \in \R_{\ge 0}: \forall x, y \in M_1: d_2 \left({f \left({x}\right), f \left({y}\right)}\right) \le L \left({d_1 \left({x, y}\right)}\right)^\alpha$ Further, $f$ is said to be '''Hölder continuous'''  it is '''$\alpha$-Hölder continuous''' for some $\alpha \in \R_{\ge 0}$.
Definition:Permutation/Ordered Selection
Let $S$ be a set of $n$ elements. Let $r \in \N: r \le n$. An '''$r$-permutation of $S$''' is an ordered selection of $r$ elements of $S$.
Definition:Subsequential Limit
Let $\sequence {x_n}$ be a sequence. Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$. Suppose that $\sequence {x_{n_r} }$ converges to a limit $x$. Then $x$ is called a '''subsequential limit''' of $\sequence {x_n}$.
Definition:Ordinal Function
Let $A$ be an ordinal (we shall allow $A$ to be a proper class). Then the mapping $G : A \to \operatorname{On}$ is called an ''ordinal function''.
Definition:Power of Element/Monoid/Invertible Element
Let $b \in S$ be invertible for $\circ$. Let $n \in \Z$. The definition $b^n = \map {\circ^n} b$ as the $n$th power of $b$ in $\left({S, \circ}\right)$ can be extended to include the inverse of $b$: :$b^{-n} = \paren {b^{-1} }^n$
Definition:Power of Element/Monoid
Let $\struct {S, \circ}$ be a monoid whose identity element is $e$. Let $a \in S$. Let $n \in \N$. The definition $a^n = \map {\circ^n} a$ as the $n$th power of $a$ in a semigroup can be extended to allow an exponent of $0$: :$a^n = \begin {cases} e & : n = 0 \\ a^{n - 1} \circ a & : n > 0 \end{cases}$ or :$n \cdot a = \begin {cases} e & : n = 0 \\ \paren {n - 1} a \circ a & : n > 0 \end{cases}$ The validity of this definition follows from the fact that a monoid has an identity element. === Invertible Element === {{:Definition:Power of Element/Monoid/Invertible Element}}
Definition:Power of Element/Group
Let $\struct {G, \circ}$ be a group whose identity element is $e$. Let $g \in G$. Let $n \in \Z$. The definition $g^n = \map {\circ^n} g$ as the $n$th power of $g$ in a monoid can be extended to allow negative values of $n$: :$g^n = \begin{cases} e & : n = 0 \\ g^{n - 1} \circ g & : n > 0 \\ \paren {g^{-n} }^{-1} & : n < 0 \end{cases}$ or :$n \cdot g = \begin{cases} e & : n = 0 \\ \paren {\paren {n - 1} \cdot g} \circ g & : n > 0 \\ -\paren {-n \cdot g} & : n < 0 \end{cases}$ The validity of this definition follows from the group axioms: $g$ has an inverse element.
Definition:Canonical Order
Let $\operatorname{On}$ be the ordinal class. Let $<$ be the ordinal ordering. Let $\operatorname{Le}$ be the lexicographic ordering on $\operatorname{On} \times \operatorname{On}$. The '''canonical order''' on $\operatorname{On} \times \operatorname{On}$, denoted $R_0$, is defined as follows, for ordinals $\alpha, \beta, \gamma, \delta$: ::$\left({\alpha, \beta}\right) \mathrel{R_0} \left({\gamma, \delta}\right)$ :: ::$\max \left({\alpha, \beta}\right) < \max \left({\gamma, \delta}\right)$ or $\left({\max \left({ \alpha, \beta}\right) = \max \left({\gamma, \delta}\right) \land \left({\alpha, \beta}\right) \mathrel{\operatorname{Le}} \left({\gamma, \delta}\right)}\right)$
Definition:Ordinal Addition
Let $x$ and $y$ be ordinals. The operation of '''ordinal addition''' $x + y$ is defined using transfinite recursion on $y$, as follows. === Base Case === When $y = \varnothing$, define: :$x + \varnothing := x$ === Inductive Case === For a successor ordinal $y^+$, define: :$x + y^+ := \left({x + y}\right)^+$ === Limit Case === Let $y$ be a limit ordinal. Then: :$\displaystyle x + y := \bigcup_{z \mathop \in y} \left({x + z}\right)$
Definition:Relational Loop
Let $\RR$ be a relation on a set $S$. Let $a_1, a_2, \ldots a_n$ be elements of $S$. A '''relational loop on $S$''' takes the form: :$\tuple {a_1 \mathrel \RR a_2 \land a_2 \mathrel \RR a_3 \dots \land a_{n - 1} \mathrel \RR a_n \land a_n \mathrel \RR a_1}$ That is, it is a subset of $\RR$ of the form: :$\set {\tuple {a_1, a_2}, \tuple {a_2, a_3}, \ldots, \tuple {a_{n - 1}, a_n}, \tuple {a_n, a_1} }$
Definition:Real Matrix
A '''real matrix''' is a matrix whose elements are all real numbers.
Definition:Smooth Homotopy
Let $X$ and $Y$ be topological spaces. Let $f: X \to Y$, $g: X \to Y$ be smooth mappings. Then $f$ and $g$ are '''smoothly homotopic'''  there exists a smooth mapping: : $H: X \times \left[{0 \,.\,.\, 1}\right] \to Y$ such that: : $H \left({x, 0}\right) = f \left({x}\right)$ and: : $H \left({x, 1}\right) = g \left({x}\right)$ $H$ is called a '''smooth homotopy between $f$ and $g$'''.
Definition:Basic Primitive Recursive Function/Zero Function
The '''zero function''' $\Zero: \N \to \N$ is a basic primitive recursive function, defined as: :$\forall n \in \N: \map \Zero n = 0$
Definition:Basic Primitive Recursive Function/Successor Function
The '''successor function''' $\Succ: \N \to \N$ is a basic primitive recursive function, defined as: :$\forall n \in \N: \map \Succ n = n + 1$
Definition:Basic Primitive Recursive Function/Projection Function
The '''projection functions''' $\pr_j^k: \N^k \to \N$ are basic primitive recursive functions, defined as: :$\forall \tuple {n_1, n_2, \ldots, n_k} \in \N^k: \map {\pr_j^k} {n_1, n_2, \ldots, n_k} = n_j$<ref>The usual notation for the projection function omits the superscript that defines the arity of the particular instance of the projection in question at the time, for example: $\pr_j$. However, in the context of computability theory, it is a ''very good idea'' to be ''completely certain'' of ''exactly'' which projection function is under discussion.</ref> where $j \in \closedint 1 k$.
Definition:Basic Primitive Recursive Function/Identity Function
The identity function $I_\N: \N \to \N$ is a basic primitive recursive function, defined as: :$\forall n \in \N: \map {I_\N} n = n$ Note that this is an implementation of the projection function: :$\pr_1^1: \N \to \N: \map {\pr_1^1} {n_1} = n_1$
Definition:Basic Primitive Recursive Function/URM Computability
The basic primitive recursive functions: * Zero Function * Successor Function * Projection Function * Identity Function are each URM computable by a single-instruction URM program.
Definition:Primitive Recursive/Set
Let $A \subseteq \N$. Then $A$ is a '''primitive recursive set''' iff its characteristic function $\chi_A$ is a primitive recursive function.
Definition:Primitive Recursive/Relation
Let $\mathcal R \subseteq \N^k$ be an $n$-ary relation on $\N^k$. Then $\mathcal R$ is a '''primitive recursive relation'''  its characteristic function $\chi_\mathcal R$ is a primitive recursive function.
Definition:Primitive Recursive/Function
A function is '''primitive recursive''' if it can be obtained from basic primitive recursive functions using the operations of substitution and primitive recursion a finite number of times.
Definition:Primitive Recursion/One Variable
Let $a \in \N$ be a natural number. Let $g: \N^2 \to \N$ be a function. Then the function $h: \N \to \N$ is '''obtained from the constant $a$ and $g$ by primitive recursion''' : :$\forall n \in \N: \map h n = \begin {cases} a & : n = 0 \\ \map g {n - 1, \map h {n - 1} } & : n > 0  \end{cases}$
Definition:Primitive Recursion/Partial Function
Let $f: \N^k \to \N$ and $g: \N^{k+2} \to \N$ be partial functions. Let $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$. Then the partial function $h: \N^{k + 1} \to \N$ is '''obtained from $f$ and $g$ by primitive recursion''' : :$\forall n \in \N: \map h {n_1, n_2, \ldots, n_k, n} \approx \begin {cases} \map f {n_1, n_2, \ldots, n_k} & : n = 0 \\ \map g {n_1, n_2, \ldots, n_k, n - 1, \map h {n_1, n_2, \ldots, n_k, n - 1} } & : n > 0  \end{cases}$ where $\approx$ is as defined in Partial Function Equality. Note that $\map h {n_1, n_2, \ldots, n_k, n}$ is defined only when: :$\map h {n_1, n_2, \ldots, n_k, n - 1}$ is defined :$\map g {n_1, n_2, \ldots, n_k, n - 1, \map h {n_1, n_2, \ldots, n_k, n - 1} }$ is defined.
Definition:Primitive Recursion/Several Variables
Let $f: \N^k \to \N$ and $g: \N^{k + 2} \to \N$ be functions. Let $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$. Then the function $h: \N^{k + 1} \to \N$ is '''obtained from $f$ and $g$ by primitive recursion''' : :$\forall n \in \N: \map h {n_1, n_2, \ldots, n_k, n} = \begin {cases} \map f {n_1, n_2, \ldots, n_k} & : n = 0 \\ \map g {n_1, n_2, \ldots, n_k, n - 1, \map h {n_1, n_2, \ldots, n_k, n - 1} } & : n > 0  \end {cases}$
Definition:Minimization/Partial Function
Let $f: \N^{k+1} \to \N$ be a partial function. Let $n = \left({n_1, n_2, \ldots, n_k}\right) \in \N^k$ be fixed. Then the '''minimization operation on $f$''' is written as: :$\mu y \left({f \left({n, y}\right) = 0}\right)$ and is specified as follows: :$\mu y \left({f \left({n, y}\right) = 0}\right) = \begin{cases} z & : f \left({n, z}\right) = 0 \text { and } f \left({n, y}\right) \text{ defined and } \forall y: 0 \le y < z: f \left({n, y}\right) \ne 0 \\ \text{undefined} & : \text{otherwise} \end{cases}$ The partial function: :$g \left({n}\right) \approx \mu y \left({f \left({n, y}\right) = 0}\right)$ obtained in this way (see Partial Function Equality) is said to be obtained from $f$ '''by minimization'''. ==== Note ==== It is not enough for there to exist $z$ such that $f \left({n, z}\right) = 0$. We need to insist that $f \left({n, y}\right)$ is actually defined for all $y$ less than $z$. Otherwise, if we were to try and find $z$ by the recursive technique of trying all $z$ from $0$ up, we would never actually get up as far as $z$ because the undefined value of $f$ for some $y$ is getting in the way. In the context of URM programs, this is significant, as an undefined output from a function is determined by a non-terminating program.
Definition:Recursive/Function
A function is '''recursive''' if it can be obtained from basic primitive recursive functions using the operations of: : substitution : primitive recursion, and  : minimization on a function a finite number of times.
Definition:Recursive/Set
Let $A \subseteq \N$. Then $A$ is a '''recursive set''' iff its characteristic function $\chi_A$ is a recursive function.
Definition:Recursive/Relation
Let $\mathcal R \subseteq \N^k$ be an $n$-ary relation on $\N^k$. Then $\mathcal R$ is a '''recursive relation'''  its characteristic function $\chi_\mathcal R$ is a recursive function.
Definition:Special Linear Group
Let $R$ be a commutative ring with unity whose zero is $0$ and unity is $1$. The '''special linear group of order $n$ on $R$''' is the set of square matrices of order $n$ whose determinant is $1$. It is a group under (conventional) matrix multiplication.
Definition:Antiassociative Operation
Let $\circ$ be a binary operation on the set $S$. $\circ$ is an '''antiassociative operation''' on $S$ : :$\forall x, y, z \in S: \left({x \circ y}\right)\circ z \ne x \circ \left({y \circ z}\right)$
Definition:Antiassociative Structure
Let $\left({S, \circ}\right)$ be an algebraic strcuture. Then $\left({S, \circ}\right)$ is an '''antiassociative structure'''  $\circ$ is an antiassociative operation. That is, : :$\forall x, y, z \in S: \left({x \circ y}\right)\circ z \ne x \circ \left({y \circ z}\right)$
Definition:Hadamard Product
Let $\struct {S, \cdot}$ be an algebraic structure. Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $S$. Let $\mathbf B = \sqbrk b_{m n}$ be an $m \times n$ matrix over $S$. The '''Hadamard product of $\mathbf A$ and $\mathbf B$''' is written $\mathbf A \circ \mathbf B$ and is defined as follows: :$\mathbf A \circ \mathbf B := \mathbf C = \sqbrk c_{m n}$ where: :$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = a_{i j} \cdot_R b_{i j}$
Definition:Magma of Sets
Let $X$ be a set. Let $\SS \subseteq \powerset X$ be a set of subsets of $X$. Let $I$ be an index set. For every $i \in I$, let $J_i$ be an index set, and let: :$\phi_i: \powerset X^{J_i} \to \powerset X$ be a partial mapping. Then $\SS$ is a '''magma of sets for $\set {\phi_i: i \in I}$ on $X$''' : :$\forall i \in I: \map {\phi_i} {\family {S_j}_{j \mathop \in J_i} } \in \SS$ for every indexed family $\family {S_j}_{j \mathop \in J_i} \in \SS^{J_i}$ in the domain of $\phi$.
Definition:Magma of Sets Generated by Collection of Subsets
Let $X$ be a set, and let $\Phi := \left\{{\phi_i: i \in I}\right\}$ be a collection of partial mappings with codomain $\mathcal P \left({X}\right)$, the power set of $X$. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then the '''magma of sets for $\Phi$ generated by $\mathcal G$''' is the unique magma of sets $\mathcal S \subseteq \mathcal P \left({X}\right)$ satisfying: :$(1): \quad \mathcal G \subseteq \mathcal S$ :$(2): \quad \mathcal G \subseteq \mathcal T$ implies that $\mathcal S \subseteq \mathcal T$ for every magma of sets $\mathcal T$ To speak of ''the unique'' '''magma of sets generated by $\mathcal G$''' is justified by Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets.
Definition:Cantor Set/Limit of Decreasing Sequence
Let $\map {I_c} \R$ denote the set of all closed real intervals. Define the mapping $t_1: \map {I_c} \R \to \map {I_c} \R$ by: :$\map {t_1} {\closedint a b} := \closedint a {\dfrac 1 3 \paren {a + b} }$ and similarly $t_3: \map {I_c} \R \to \map {I_c} \R$ by: :$\map {t_3} {\closedint a b} := \closedint {\dfrac 2 3 \paren {a + b} } b$ Note in particular how: :$\map {t_1} {\closedint a b} \subseteq \closedint a b$ :$\map {t_3} {\closedint a b} \subseteq \closedint a b$ Subsequently, define inductively: :$S_0 := \set {\closedint 0 1}$ :$S_{n + 1} := \map {t_1} {C_n} \cup \map {t_3} {C_n}$ and put, for all $n \in \N$: :$C_n := \displaystyle \bigcup S_n$ Note that $C_{n + 1} \subseteq C_n$ for all $n \in \N$, so that this forms a decreasing sequence of sets. Then the '''Cantor set''' $\mathcal C$ is defined as its limit, that is: :$\mathcal C := \displaystyle \bigcap_{n \mathop \in \N} C_n$
Definition:Ordinal Subtraction
Let $x$ and $y$ be ordinals such that $x \le y$. Then the operation of '''ordinal subtraction''' is defined as: :$\displaystyle y - x = \bigcup \left\{{z: x + z = y}\right\}$
Definition:Ordinal Multiplication
Let $x$ and $y$ be ordinals. The operation of '''ordinal multiplication''' $x \times y$ is defined using transfinite recursion as follows: === Base Case === {{begin-eqn}} {{eqn | l = \left({x \times \varnothing}\right)       | r = \varnothing       | c = if $y$ is equal to $\varnothing$ }} {{end-eqn}} === Inductive Case === {{begin-eqn}} {{eqn | l = \left({x \times z^+}\right)       | r = \left({x \times z}\right) + x       | c = if $y$ is the successor of some ordinal $z$ }} {{end-eqn}} === Limit Case === {{begin-eqn}} {{eqn | l = \left({x \times y}\right)       | r = \bigcup_{z \mathop \in y} \left({x \times z}\right)       | c = if $y$ is a limit ordinal }} {{end-eqn}}
Definition:Ordinal Exponentiation
Let $x$ and $y$ be ordinals. '''Ordinal exponentiation''' $x^y$ is defined using Transfinite Recursion: :$\displaystyle x^y = \begin{cases}  0 & : x = 0, \ y \ne 0 \\  & \\ 1 & : x = 0, \ y = 0 \\  & \\ 1 & : x \ne 0, \ y = 0 \\  & \\ \left({x^z \cdot x}\right) & : x \ne 0, \ y = z^+ \\  & \\ \bigcup_{z \mathop \in y} x^z & : x \ne 0, \ y \in K_{II} \\ \end{cases}$ where: * $K_{II}$ is the class of all limit ordinals * $0$ denotes the zero ordinal * $1$ denotes the ordinal $1$, that is: $0^+$, the successor of $0$.
Definition:Ordinal Sum
Let $y$ be an ordinal. Let $\left\langle{A_x}\right\rangle$ be a sequence of ordinals. The '''ordinal sum of $A_x$''' is denoted $\displaystyle \sum_{x \mathop = 1}^y A_x$ and defined using Transfinite Recursion on $y$ as follows: :$\displaystyle \sum_{x \mathop = 1}^\varnothing A_x = \varnothing$ :$\displaystyle \sum_{x \mathop = 1}^{z^+} A_x = \sum_{x \mathop = 1}^z \left({A_x}\right) + A_{z^+}$ :$\displaystyle \sum_{x \mathop = 1}^y A_x = \bigcup_{z \mathop \in y} \left({\sum_{x \mathop = 1}^z A_x}\right)$ for limit ordinals $y$.
Definition:Anticommutative/Structure with One Operation
Let $\left({S, \circ}\right)$ be an algebraic structure Then $\circ$ is '''anticommutative on $S$''' : :$\forall x, y \in S: x \circ y = y \circ x \iff x = y$ Equivalently, it can be defined as: :$\forall x, y \in S: x \ne y \iff x \circ y \ne y \circ x$
Definition:Anticommutative/Structure with Two Operations
Let $\left({S, +, \circ}\right)$ be an algebraic structure. Suppose every element $x$ in $\left({S, +}\right)$ has an inverse element $-x$. Then $\circ$ is '''anticommutative on $S$ with respect to $+$''' : :$\forall x, y \in S: x \circ y = -\left({y \circ x}\right)$
Definition:Middle Cancellation Property
Let $\left({S, \circ}\right)$ be a semigroup. Then $\circ$ is said to have the '''middle cancellation property''' iff: :$a \circ x \circ b = c \circ x \circ d \implies a \circ b = c \circ d$ holds for all $a,b,c,d,x \in S$.
Definition:Cross Cancellation Property
Let $\left({S, \circ}\right)$ be an algebraic structure. Then $\circ$ is said to have the '''cross cancellation property''' iff: :$a \circ b = c \circ a \implies b = c$ holds for all $a, b, c \in S$.
Definition:Mutatis Mutandis
'''Mutatis mutandis''' is Latin for '''what needs changing having been changed'''. It can be used in place of words like ''analogously'' and ''similarly'', and expresses that an argument can be applied to a new situation in an intuitive fashion. For example the proof of Union is Associative can be made into that for Intersection is Associative by '''mutatis mutandis'''. In this case, this comes down to replacing $\cup$ by $\cap$, and $\lor$ by $\land$. In particular, the essential flow of the argument was not changed in adapting the proof.
Definition:Uniformly Integrable
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\mathcal F \subseteq \mathcal M \left({\Sigma}\right)$ be a collection of measurable functions. Then $\mathcal F$ is said to be '''uniformly integrable (with respect to $\mu$)''' iff: :$\forall \epsilon > 0: \exists g_\epsilon \in \mathcal{L}^1_+ \left({\mu}\right): \displaystyle \sup_{f \mathop \in \mathcal F} \int_{\left\{{\left\vert{f}\right\vert > g_\epsilon}\right\}} \left\vert{f}\right\vert \, \mathrm d \mu < \epsilon$ where: :$\mathcal{L}^1_+ \left({\mu}\right)$ is the space of positive $\mu$-integrable functions :$\left\{{\left\vert{f}\right\vert > g_\epsilon}\right\}$ is short for $\left\{{x \in X: \left\vert{f \left({x}\right)}\right\vert > g_\epsilon \left({x}\right)}\right\}$
Definition:Structure (Set Theory)
Let $A$ be a class. Let $\mathcal R$ be a relation. The '''relational structure $\left[{A, \mathcal R}\right]$ satisfies well-formed formula $p$''', denoted $\left[{A, \mathcal R}\right] \models p$, shall be defined on the well-formed parts of $p$: {{begin-eqn}} {{eqn | l = \left[{A, \mathcal R}\right] \models x \in y       | o = \iff       | r = \left({x \in A \land y \in A \land x \mathrel {\mathcal R} y}\right) }} {{eqn | l = \left[{A, \mathcal R}\right] \models \neg p       | o = \iff       | r = \neg \left[{A, \mathcal R}\right] \models p }} {{eqn | l = \left[{A, \mathcal R}\right] \models \left({p \land q}\right)       | o = \iff       | r = \left({\left[{A, \mathcal R}\right] \models p \land \left[{A, \mathcal R}\right] \models q}\right) }} {{eqn | l = \left[{A, \mathcal R}\right] \models \forall x: P \left({x}\right)       | o = \iff       | r = \forall x \in A: \left[{A, \mathcal R}\right] \models P \left({x}\right) }} {{end-eqn}}
Definition:Standard Structure
The structure $\left[{A,R}\right]$ is called a ''standard structure'' iff: :$\displaystyle R = \Epsilon \cap \left({ A \times A }\right)$ where $\Epsilon$ denotes the epsilon relation and $\times$ denotes the Cartesian product. With a standard structure, $\left[{A,R}\right] \models p$ shall be abbreviated: :$\displaystyle A \models p \iff \left[{A , E \cap \left({ A \times A }\right)}\right] \models p$
Definition:Relativisation
Let $p$ be a well-formed formula of the language of set theory. Let $A$ be a class. The '''relativisation of $p$ to $A$''' shall be denoted $p^A$ and shall be defined recursively on the symbols in $p$: :$x \in y ^A \iff x \in y$ :$\left({\neg p}\right)^A \iff \neg p^A$ :$\left({p \land q}\right)^A \iff \left({p^A \land q^A}\right)$ :$\left({\forall x: P \left({x}\right)}\right)^A \iff \forall x: \left({ x \in A \implies P \left({x}\right)^A}\right)$ Thus, the relativisation of $p$ is simply the well-formed formula achieved when replacing all instances of $\forall x$ with $\forall x \in A$.
Definition:Restricted Universal Quantifier
Let $A$ be a class in ZF. The '''restricted universal quantifier''' is denoted $\forall x \in A$ and is defined as the following definitional abbreviation: :$\forall x \in A: P \left({x}\right) \quad \text{for} \quad \forall x: \left({x \in A \implies P \left({x}\right)}\right)$ where $P \left({x}\right)$ is any well-formed formula of the language of set theory.