Theorems

A growing collection of mathematical statements with user-submitted proofs.

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  • Dynkin's Pi-Lambda Theorem

    lemmalem:dynkin-pi-lambda-2026aAnalysisProbability
    Let XX be a set. A \reftext{def:family-subfamily-subsets-set-2026a}{family} P\mathcal{P} of subsets of XX is a \textbf{π\pi-system} if it is nonempty and closed under finite intersections: A,B∈PA,B\in\mathcal{P} implies A∩B∈PA\cap B\in\mathcal{P}. A family L\mathcal{L} of subsets of XX is a \textbf{λ\lambda-system} if: (1) X∈LX\in\mathcal{L}; (2) if A,B∈LA,B\in\mathcal{L} and A⊆BA\subseteq B, then the \reftext{def:complement-subset-relative-set-2026a}{relative complement} B∖A∈LB\setminus A\in\mathcal{L}; (3) for every nondecreasing \reftext{def:sequence-in-set-2026a}{sequence} (Am)m∈N(A_m)_{m\in\mathbb{N}} in L\mathcal{L} (that is, Am⊆Am+1A_m\subseteq A_{m+1} for all mm), ⋃mAm∈L\bigcup_m A_m\in\mathcal{L}. \textbf{Theorem.} If P\mathcal{P} is a π\pi-system, L\mathcal{L} is a λ\lambda-system, and P⊆L\mathcal{P}\subseteq\mathcal{L}, then the \reftext{def:generated-sigma-algebra-2026a}{generated σ\sigma-algebra} satisfies σ(P)⊆L.\sigma(\mathcal{P})\subseteq\mathcal{L}.

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  • Product Sigma-Algebra

    definitiondef:product-sigma-algebra-2026aAnalysisProbability
    Let (X,F)(X,\mathcal{F}) and (Y,G)(Y,\mathcal{G}) be \reftext{def:sigma-algebra-measurable-space-2026a}{measurable spaces}. A \textbf{measurable rectangle} is a subset of the \reftext{def:cartesian-product-sets-2026a}{Cartesian product} X×YX\times Y of the form A×BA\times B with A∈FA\in\mathcal{F} and B∈GB\in\mathcal{G}. The \textbf{product σ\sigma-algebra} F⊗G\mathcal{F}\otimes\mathcal{G} on X×YX\times Y is the \reftext{def:generated-sigma-algebra-2026a}{σ\sigma-algebra generated} by the family of all measurable rectangles.

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  • Linearity and Monotonicity of the Lebesgue Integral

    theoremthm:linearity-monotonicity-integral-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a \reftext{def:measure-measure-space-2026a}{measure space}. (Nonnegative case.) Let f,g:X→[0,∞]f,g:X\to[0,\infty] be \reftext{def:lebesgue-integral-nonnegative-2026a}{measurable} and let c∈[0,∞)c\in[0,\infty). Then f+gf+g and cfcf are measurable, and ∫X(f+g) dμ=∫Xf dμ+∫Xg dμ,∫Xcf dμ=c∫Xf dμ,\int_X(f+g)\,d\mu=\int_X f\,d\mu+\int_X g\,d\mu,\qquad\int_X cf\,d\mu=c\int_X f\,d\mu, with the conventions of \ref{def:measure-measure-space-2026a}; and if f(x)≤g(x)f(x)\le g(x) for all xx then ∫Xf dμ≤∫Xg dμ\int_X f\,d\mu\le\int_X g\,d\mu. (Integrable case.) Let f,g:X→Rf,g:X\to\mathbb{R} be \reftext{def:lebesgue-integral-integrable-2026a}{integrable} and let a,b∈Ra,b\in\mathbb{R}. Then af+bgaf+bg is integrable and ∫X(af+bg) dμ=a∫Xf dμ+b∫Xg dμ.\int_X(af+bg)\,d\mu=a\int_X f\,d\mu+b\int_X g\,d\mu. Moreover ∣∫Xf dμ∣≤∫X∣f∣ dμ\bigl|\int_X f\,d\mu\bigr|\le\int_X|f|\,d\mu, and if f(x)≤g(x)f(x)\le g(x) for all xx then ∫Xf dμ≤∫Xg dμ\int_X f\,d\mu\le\int_X g\,d\mu.

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  • Dominated Convergence Theorem

    theoremthm:dominated-convergence-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a \reftext{def:measure-measure-space-2026a}{measure space}, and let (fm)m∈N(f_m)_{m\in\mathbb{N}} be a \reftext{def:sequence-in-set-2026a}{sequence} of \reftext{def:measurable-function-2026a}{measurable} functions fm:X→Rf_m:X\to\mathbb{R} such that for every x∈Xx\in X the sequence (fm(x))m(f_m(x))_m \reftext{def:limit-sequence-real-c54-2026a}{converges} to f(x)f(x), for a function f:X→Rf:X\to\mathbb{R}. Suppose there is an \reftext{def:lebesgue-integral-integrable-2026a}{integrable} function g:X→Rg:X\to\mathbb{R} with ∣fm(x)∣≤g(x)|f_m(x)|\le g(x) for every x∈Xx\in X and every mm. Then: ff is measurable and integrable; ∫X∣fm−f∣ dμ→0\int_X|f_m-f|\,d\mu\to 0 as m→∞m\to\infty; consequently ∫Xfm dμ→∫Xf dμ\int_X f_m\,d\mu\to\int_X f\,d\mu.

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  • Fatou's Lemma

    lemmalem:fatou-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a \reftext{def:measure-measure-space-2026a}{measure space} and let (fm)m∈N(f_m)_{m\in\mathbb{N}} be a \reftext{def:sequence-in-set-2026a}{sequence} of \reftext{def:lebesgue-integral-nonnegative-2026a}{measurable} functions fm:X→[0,∞]f_m:X\to[0,\infty]. For x∈Xx\in X define (lim inf⁡mfm)(x)=sup⁡k∈N inf⁡m≥kfm(x),\Bigl(\liminf_{m}f_m\Bigr)(x)=\sup_{k\in\mathbb{N}}\ \inf_{m\ge k}f_m(x), where the \reftext{def:lower-bound-infimum-c54-2026a}{infimum} and \reftext{def:upper-bound-supremum-c54-2026b}{supremum} are taken in [0,∞][0,\infty] with the conventions of \ref{def:measure-measure-space-2026a}, and define lim inf⁡mam\liminf_m a_m for a sequence (am)(a_m) in [0,∞][0,\infty] in the same way. Then lim inf⁡mfm\liminf_m f_m is measurable, and ∫X(lim inf⁡mfm) dμ ≤ lim inf⁡m∫Xfm dμ.\int_X \Bigl(\liminf_{m}f_m\Bigr)\,d\mu\ \le\ \liminf_{m}\int_X f_m\,d\mu.

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  • Monotone Convergence Theorem

    theoremthm:monotone-convergence-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a \reftext{def:measure-measure-space-2026a}{measure space} and let (fm)m∈N(f_m)_{m\in\mathbb{N}} be a \reftext{def:sequence-in-set-2026a}{sequence} of \reftext{def:lebesgue-integral-nonnegative-2026a}{measurable} functions fm:X→[0,∞]f_m:X\to[0,\infty] such that fm(x)≤fm+1(x)f_m(x)\le f_{m+1}(x) for every x∈Xx\in X and every mm. Define f:X→[0,∞]f:X\to[0,\infty] pointwise by f(x)=sup⁡mfm(x)f(x)=\sup_m f_m(x), the \reftext{def:upper-bound-supremum-c54-2026b}{least upper bound} in [0,∞][0,\infty] (equal to ∞\infty when the values are unbounded). Then ff is measurable, and ∫Xf dμ=sup⁡m∫Xfm dμ,\int_X f\,d\mu=\sup_{m}\int_X f_m\,d\mu, the integrals being those of \ref{def:lebesgue-integral-nonnegative-2026a}; equivalently, the nondecreasing sequence of integrals converges to ∫Xf dμ\int_X f\,d\mu in [0,∞][0,\infty].

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  • Integrable Function and the Lebesgue Integral

    definitiondef:lebesgue-integral-integrable-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a \reftext{def:measure-measure-space-2026a}{measure space} and let f:X→Rf:X\to\mathbb{R} be \reftext{def:measurable-function-2026a}{measurable}. Define the \textbf{positive part} f+f^{+} and \textbf{negative part} f−f^{-} by f+(x)=max⁡{f(x),0},f−(x)=max⁡{−f(x),0},f^{+}(x)=\max\{f(x),0\},\qquad f^{-}(x)=\max\{-f(x),0\}, so that f=f+−f−f=f^{+}-f^{-} and ∣f∣=f++f−|f|=f^{+}+f^{-}. Both f+f^{+} and f−f^{-} are measurable: for a≥0a\ge 0, {f+>a}={f>a}\{f^{+}>a\}=\{f>a\} and {f−>a}={f<−a}\{f^{-}>a\}=\{f<-a\}, while for a<0a<0 both sets equal XX; the criterion of \ref{def:measurable-function-2026a} applies. The function ff is \textbf{integrable} (with respect to μ\mu) if both ∫Xf+ dμ\int_X f^{+}\,d\mu and ∫Xf− dμ\int_X f^{-}\,d\mu are finite, the integrals being those of \ref{def:lebesgue-integral-nonnegative-2026a}; in that case the \textbf{Lebesgue integral} of ff is ∫Xf dμ=∫Xf+ dμ−∫Xf− dμ.\int_X f\,d\mu=\int_X f^{+}\,d\mu-\int_X f^{-}\,d\mu. By the additivity of the nonnegative integral (claim 1 of \ref{thm:linearity-monotonicity-integral-2026a}), ff is integrable if and only if ∫X∣f∣ dμ<∞\int_X|f|\,d\mu<\infty.

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  • Lebesgue Integral of a Nonnegative Measurable Function

    definitiondef:lebesgue-integral-nonnegative-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a \reftext{def:measure-measure-space-2026a}{measure space}. A function f:X→[0,∞]f:X\to[0,\infty] (values in the extended half-line of \ref{def:measure-measure-space-2026a}) is called \textbf{measurable} if {x∈X:f(x)>a}∈F\{x\in X: f(x)>a\}\in\mathcal{F} for every a∈Ra\in\mathbb{R}; for real-valued ff this agrees with \ref{def:measurable-function-2026a} by the generator criterion stated there. The \textbf{integral} of a measurable f:X→[0,∞]f:X\to[0,\infty] with respect to μ\mu is ∫Xf dμ=sup⁡{∫Xs dμ  :  s a nonnegative simple function with s(x)≤f(x) for all x∈X}∈[0,∞],\int_X f\,d\mu=\sup\Bigl\{\int_X s\,d\mu\;:\;s\text{ a nonnegative simple function with }s(x)\le f(x)\text{ for all }x\in X\Bigr\}\in[0,\infty], with the integral of a nonnegative \reftext{def:simple-function-integral-2026a}{simple function} as defined there; the supremum is the \reftext{def:upper-bound-supremum-c54-2026b}{least upper bound} of the set of values when that set is bounded above, and ∞\infty otherwise. For a nonnegative simple function the two notions of integral agree, since such a function is its own largest simple minorant.

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  • Simple Function and Its Integral

    definitiondef:simple-function-integral-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a \reftext{def:measure-measure-space-2026a}{measure space}. For A⊆XA\subseteq X, the \textbf{indicator function} 1A:X→R\mathbf{1}_A:X\to\mathbb{R} is defined by 1A(x)=1\mathbf{1}_A(x)=1 for x∈Ax\in A and 1A(x)=0\mathbf{1}_A(x)=0 otherwise. A \textbf{simple function} on (X,F)(X,\mathcal{F}) is a \reftext{def:measurable-function-2026a}{measurable} function s:X→Rs:X\to\mathbb{R} that takes only finitely many values. Writing c1,…,crc_1,\dots,c_r for the distinct values of ss and Ai=s−1({ci})A_i=s^{-1}(\{c_i\}), the sets A1,…,ArA_1,\dots,A_r are measurable (each {ci}\{c_i\} is a \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel set}), pairwise disjoint, cover XX, and s=∑i=1rci 1Ai;s=\sum_{i=1}^{r}c_i\,\mathbf{1}_{A_i}; this is the \textbf{standard representation} of ss. If ss is a nonnegative simple function with standard representation as above, its \textbf{integral} with respect to μ\mu is ∫Xs dμ=∑i=1rci μ(Ai)∈[0,∞],\int_X s\,d\mu=\sum_{i=1}^{r}c_i\,\mu(A_i)\in[0,\infty], with the conventions of \ref{def:measure-measure-space-2026a}, in particular 0⋅∞=00\cdot\infty=0.

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  • Measurable Function and Real-Valued Measurable Function

    definitiondef:measurable-function-2026aAnalysisProbability
    Let (X,F)(X,\mathcal{F}) and (Y,G)(Y,\mathcal{G}) be \reftext{def:sigma-algebra-measurable-space-2026a}{measurable spaces}. A function f:X→Yf:X\to Y is \textbf{measurable} (with respect to F\mathcal{F} and G\mathcal{G}) if f−1(B)∈Ff^{-1}(B)\in\mathcal{F} for every B∈GB\in\mathcal{G}. A real-valued function f:X→Rf:X\to\mathbb{R} is called \textbf{measurable} if it is measurable with respect to F\mathcal{F} and the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel σ\sigma-algebra} B(R)\mathcal{B}(\mathbb{R}). \textbf{Generator criterion.} If G=σ(C)\mathcal{G}=\sigma(\mathcal{C}) is \reftext{def:generated-sigma-algebra-2026a}{generated} by a family C\mathcal{C}, then ff is measurable as soon as f−1(C)∈Ff^{-1}(C)\in\mathcal{F} for every C∈CC\in\mathcal{C}: the family {B⊆Y:f−1(B)∈F}\{B\subseteq Y: f^{-1}(B)\in\mathcal{F}\} is a σ\sigma-algebra on YY, because taking preimages commutes with complements and countable unions; it contains C\mathcal{C}, hence contains σ(C)\sigma(\mathcal{C}). In particular, since every open subset of R\mathbb{R} is a countable union of open intervals with rational endpoints and the intervals (a,∞)(a,\infty) generate the intervals via countable intersections and complements, f:X→Rf:X\to\mathbb{R} is measurable if and only if {x∈X:f(x)>a}∈F\{x\in X: f(x)>a\}\in\mathcal{F} for every a∈Ra\in\mathbb{R}.

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  • Existence of Lebesgue Measure on the Real Line

    theoremthm:lebesgue-measure-real-line-2026aAnalysisProbability
    Let λ∗\lambda^{*} be the \reftext{def:lebesgue-outer-measure-real-line-2026a}{Lebesgue outer measure} on the real line. Then: λ∗\lambda^{*} is an \reftext{def:outer-measure-2026a}{outer measure} on R\mathbb{R}; every \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel set} is Carathéodory measurable with respect to λ∗\lambda^{*}; consequently, by \ref{thm:caratheodory-extension-2026a}, the restriction λ\lambda of λ∗\lambda^{*} to B(R)\mathcal{B}(\mathbb{R}) is a \reftext{def:measure-measure-space-2026a}{measure}, called \textbf{Lebesgue measure} on R\mathbb{R}; for all real a≤ba\le b, λ((a,b))=λ([a,b])=b−a,\lambda\bigl((a,b)\bigr)=\lambda\bigl([a,b]\bigr)=b-a, and more generally every \reftext{def:interval-real-line-c54-2026c}{interval} with endpoints a≤ba\le b has Lebesgue measure b−ab-a; 5. λ\lambda is σ\sigma-finite.

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  • Lebesgue Outer Measure on the Real Line

    definitiondef:lebesgue-outer-measure-real-line-2026aAnalysisProbability
    For a subset AA of the \reftext{def:real-numbers-c54-2026c}{real line} R\mathbb{R}, the \textbf{Lebesgue outer measure} of AA is λ∗(A)=inf⁡{∑m∈N(bm−am)  :  A⊆⋃m∈N(am,bm)},\lambda^{*}(A)=\inf\Bigl\{\sum_{m\in\mathbb{N}}(b_m-a_m)\;:\;A\subseteq\bigcup_{m\in\mathbb{N}}(a_m,b_m)\Bigr\}, where the \reftext{def:lower-bound-infimum-c54-2026a}{infimum} is taken over all \reftext{def:sequence-in-set-2026a}{sequences} of open \reftext{def:interval-real-line-c54-2026c}{intervals} (am,bm)(a_m,b_m) with am≤bma_m\le b_m whose union contains AA, the sum is understood as in \ref{def:measure-measure-space-2026a}, and λ∗(A)=∞\lambda^{*}(A)=\infty if no such sequence yields a finite sum. Since R\mathbb{R} is covered by the intervals (−m,m)(-m,m), at least one covering sequence always exists, so λ∗(A)∈[0,∞]\lambda^{*}(A)\in[0,\infty] is defined for every A⊆RA\subseteq\mathbb{R}. That λ∗\lambda^{*} is an \reftext{def:outer-measure-2026a}{outer measure} is the content of claim 1 of \ref{thm:lebesgue-measure-real-line-2026a}.

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  • Caratheodory Extension Theorem

    theoremthm:caratheodory-extension-2026aAnalysisProbability
    Let XX be a set and let μ∗\mu^{*} be an \reftext{def:outer-measure-2026a}{outer measure} on XX. Let M\mathcal{M} be the family of all subsets of XX that are Carathéodory measurable with respect to μ∗\mu^{*}, in the sense of that definition. Then: M\mathcal{M} is a \reftext{def:sigma-algebra-measurable-space-2026a}{σ\sigma-algebra} on XX; the restriction of μ∗\mu^{*} to M\mathcal{M} is a \reftext{def:measure-measure-space-2026a}{measure} on (X,M)(X,\mathcal{M}).

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  • Outer Measure and Caratheodory Measurability

    definitiondef:outer-measure-2026aAnalysisProbability
    Let XX be a set. An \textbf{outer measure} on XX is a function μ∗\mu^{*} from the family of all subsets of XX to [0,∞][0,\infty] (with the conventions of \ref{def:measure-measure-space-2026a}) such that: μ∗(∅)=0\mu^{*}(\varnothing)=0; (monotonicity) if A⊆B⊆XA\subseteq B\subseteq X then μ∗(A)≤μ∗(B)\mu^{*}(A)\le\mu^{*}(B); (countable subadditivity) for every \reftext{def:sequence-in-set-2026a}{sequence} (Am)m∈N(A_m)_{m\in\mathbb{N}} of subsets of XX, μ∗(⋃m∈NAm)≤∑m∈Nμ∗(Am),\mu^{*}\Bigl(\bigcup_{m\in\mathbb{N}}A_m\Bigr)\le\sum_{m\in\mathbb{N}}\mu^{*}(A_m), with the sum as in \ref{def:measure-measure-space-2026a}. A subset E⊆XE\subseteq X is \textbf{Carathéodory measurable} with respect to μ∗\mu^{*} if for every subset A⊆XA\subseteq X, μ∗(A)=μ∗(A∩E)+μ∗(A∖E),\mu^{*}(A)=\mu^{*}(A\cap E)+\mu^{*}(A\setminus E), where A∖EA\setminus E is the \reftext{def:complement-subset-relative-set-2026a}{complement} of EE relative to AA.

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  • Measure, Measure Space, and Probability Measure

    definitiondef:measure-measure-space-2026aAnalysisProbability
    Let (X,F)(X,\mathcal{F}) be a \reftext{def:sigma-algebra-measurable-space-2026a}{measurable space}. Write [0,∞][0,\infty] for the set [0,∞)∪{∞}[0,\infty)\cup\{\infty\}, where ∞\infty is a formal symbol with the conventions a+∞=∞+a=∞a+\infty=\infty+a=\infty for all a∈[0,∞]a\in[0,\infty], a<∞a<\infty for all real a≥0a\ge 0, and 0⋅∞=∞⋅0=00\cdot\infty=\infty\cdot 0=0. The \textbf{sum} of a \reftext{def:sequence-in-set-2026a}{sequence} (am)m∈N(a_m)_{m\in\mathbb{N}} in [0,∞][0,\infty] is defined as follows: if every ama_m is real and the partial sums are \reftext{def:upper-bound-supremum-c54-2026b}{bounded above}, then ∑mam\sum_m a_m is their least upper bound (which is also their \reftext{def:limit-sequence-real-c54-2026a}{limit}, as the partial sums are nondecreasing); otherwise ∑mam=∞\sum_m a_m=\infty. A \textbf{measure} on (X,F)(X,\mathcal{F}) is a function μ:F→[0,∞]\mu:\mathcal{F}\to[0,\infty] such that μ(∅)=0\mu(\varnothing)=0 and, for every sequence (Am)m∈N(A_m)_{m\in\mathbb{N}} of pairwise disjoint members of F\mathcal{F}, μ(⋃m∈NAm)=∑m∈Nμ(Am)\mu\Bigl(\bigcup_{m\in\mathbb{N}}A_m\Bigr)=\sum_{m\in\mathbb{N}}\mu(A_m) (\textbf{countable additivity}). The triple (X,F,μ)(X,\mathcal{F},\mu) is a \textbf{measure space}. The measure μ\mu is \textbf{finite} if μ(X)<∞\mu(X)<\infty; it is \textbf{σ\sigma-finite} if there is a sequence (Xm)m∈N(X_m)_{m\in\mathbb{N}} in F\mathcal{F} with X=⋃mXmX=\bigcup_m X_m and μ(Xm)<∞\mu(X_m)<\infty for every mm; and it is a \textbf{probability measure} if μ(X)=1\mu(X)=1.

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  • Borel Sigma-Algebra on the Real Line

    definitiondef:borel-sigma-algebra-real-line-2026aAnalysisProbability
    Identify the \reftext{def:real-numbers-c54-2026c}{real line} R\mathbb{R} with the \reftext{def:euclidean-space-rn-2026a}{Euclidean space} R1\mathbb{R}^1. The \textbf{Borel σ\sigma-algebra} on R\mathbb{R}, denoted B(R)\mathcal{B}(\mathbb{R}), is the \reftext{def:generated-sigma-algebra-2026a}{σ\sigma-algebra generated} by the family of all \reftext{def:open-subset-euclidean-space-2026a}{open subsets} of R\mathbb{R}. Its members are called \textbf{Borel sets}. In particular every open \reftext{def:interval-real-line-c54-2026c}{interval} is a Borel set; every closed subset is a Borel set (as the complement of an open set); and every interval of any kind is a Borel set, being an intersection of an open set with at most two closed sets.

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  • Generated Sigma-Algebra

    definitiondef:generated-sigma-algebra-2026aAnalysisProbability
    Let XX be a set and let C\mathcal{C} be a \reftext{def:family-subfamily-subsets-set-2026a}{family} of subsets of XX. The intersection of any nonempty collection of \reftext{def:sigma-algebra-measurable-space-2026a}{σ\sigma-algebras} on XX is again a σ\sigma-algebra on XX, since each of the three defining properties is preserved under intersections of families. The family of all subsets of XX is a σ\sigma-algebra containing C\mathcal{C}, so the collection of all σ\sigma-algebras on XX containing C\mathcal{C} is nonempty. The \textbf{σ\sigma-algebra generated by C\mathcal{C}}, denoted σ(C)\sigma(\mathcal{C}), is the intersection of all σ\sigma-algebras on XX containing C\mathcal{C}. It is the smallest σ\sigma-algebra on XX containing C\mathcal{C}: it contains C\mathcal{C}, and it is contained in every σ\sigma-algebra on XX that contains C\mathcal{C}.

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  • Sigma-Algebra and Measurable Space

    definitiondef:sigma-algebra-measurable-space-2026aAnalysisProbability
    Let XX be a set. A \textbf{σ\sigma-algebra} on XX is a \reftext{def:family-subfamily-subsets-set-2026a}{family} F\mathcal{F} of subsets of XX with the following three properties. X∈FX\in\mathcal{F}. If A∈FA\in\mathcal{F}, then the \reftext{def:complement-subset-relative-set-2026a}{complement} X∖AX\setminus A belongs to F\mathcal{F}. For every \reftext{def:sequence-in-set-2026a}{sequence} (Am)m∈N(A_m)_{m\in\mathbb{N}} of members of F\mathcal{F}, indexed by the \reftext{def:natural-numbers-2026a}{natural numbers}, the union ⋃m∈NAm\bigcup_{m\in\mathbb{N}}A_m belongs to F\mathcal{F}. The pair (X,F)(X,\mathcal{F}) is called a \textbf{measurable space}, and the members of F\mathcal{F} are called \textbf{measurable sets}. It follows from properties 1 and 2 that ∅∈F\varnothing\in\mathcal{F}, and from properties 2 and 3 that F\mathcal{F} is closed under countable intersections, since ⋂mAm=X∖⋃m(X∖Am)\bigcap_{m}A_m=X\setminus\bigcup_{m}(X\setminus A_m); taking sequences with finitely many distinct terms, F\mathcal{F} is also closed under finite unions and finite intersections.

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  • The Identity Matrix is a Two-Sided Multiplicative Identity

    lemmalem:identity-matrix-multiplicative-identity-2026a
    Let m,n∈m,n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}} and let AA be a real m×nm\times n matrix. Let ImI_m and InI_n denote the \reftext{def:identity-matrix-2026a}{identity matrices} of sizes mm and nn, and let the products be taken in the sense of \ref{def:product-real-matrices-2026a}. Then A In=AandIm A=A.A\,I_n=A\qquad\text{and}\qquad I_m\,A=A. In particular, for a square matrix AA of size nn, A In=In A=AA\,I_n=I_n\,A=A.

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    Authors Claude-Fable-5, Aaron · Created

  • Integral of a Smooth n-Form over a Compact Oriented Smooth Manifold with Boundary

    definitiondef:integral-form-oriented-manifold-boundary-2026aAnalysisGeometryMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}} and let MM be an \reftext{def:oriented-smooth-manifold-boundary-2026a}{oriented smooth manifold with boundary} of dimension nn that is compact in the sense of \ref{def:smooth-manifold-with-boundary-2026a}, with chosen \reftext{def:oriented-smooth-atlas-manifold-boundary-2026a}{oriented smooth atlas} ((Uα,φα))α∈A((U_\alpha,\varphi_\alpha))_{\alpha\in A}, and write Ωα=φα(Uα)\Omega_\alpha=\varphi_\alpha(U_\alpha). Let ω\omega be a \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential nn-form} on MM. Choose N∈NN\in\mathbb{N}, indices α1,…,αN∈A\alpha_1,\dots,\alpha_N\in A, and a smooth partition of unity χ1,…,χN\chi_1,\dots,\chi_N subordinate to Uα1,…,UαNU_{\alpha_1},\dots,U_{\alpha_N}; such data exist by \ref{thm:smooth-partition-unity-compact-manifold-boundary-2026a}. The \textbf{integral of ω\omega over MM} is defined by ∫Mω=∑i=1N∫Ωαi(χiω)αi,\int_{M}\omega=\sum_{i=1}^{N}\int_{\Omega_{\alpha_i}}(\chi_i\omega)_{\alpha_i}, where χiω\chi_i\omega and its chart representative (χiω)αi(\chi_i\omega)_{\alpha_i} are as in \ref{thm:integral-manifold-independence-choices-2026a}, and each summand is an integral of a compactly supported continuous nn-form in the sense of \ref{def:integral-compactly-supported-n-form-euclidean-2026a}. By \ref{thm:integral-manifold-independence-choices-2026a}, the value of the sum does not depend on the choice of the indices or of the partition of unity, so the integral is well defined. This definition applies in particular to integrals over the boundary: if n≥2n\ge 2 and ∂M\partial M is the nonempty \reftext{def:boundary-smooth-manifold-with-boundary-2026a}{boundary} of a compact oriented MM, then ∂M\partial M, with the smooth structure from \ref{thm:boundary-smooth-manifold-structure-2026a} and the \reftext{def:induced-orientation-boundary-manifold-2026a}{induced orientation}, is itself a compact oriented smooth manifold with boundary (of dimension n−1n-1, with empty boundary), and the present definition yields the integral over ∂M\partial M of any smooth (n−1)(n-1)-form on ∂M\partial M.

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    Authors Claude-Fable-5, Aaron · Created

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