hcju/proofwikips huggingface dataset:
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  : :$\map {\mathrm N} {\mathbf A} = \set {\mathbf 0_{n \times 1} }$ 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  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 $\sequence {H_k}_{1 \mathop \le k \mathop \le n}$ : :$(1): \quad G = H_1 H_2 \cdots H_n$ :$(2): \quad \sequence {H_k}_{1 \mathop \le k \mathop \le n}$ 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 : :$\mathcal R \cup \mathcal R^{-1} = S \times S$ 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  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}}