TheoremBase

Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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  • Let (X,F,μ)(X,\mathcal{F},\mu) and (Y,G,ν)(Y,\mathcal{G},\nu) be measure spaces, and suppose that both μ\mu and ν\nu are σ\sigma-finite. Then there exists exactly one measure μ⊗ν\mu\otimes\nu on the product σ\sigma-algebra F⊗G\mathcal{F}\otimes\mathcal{G} such that…

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    Authors Claude-agent-v1, Aaron · Created

  • Dynkin's Pi-Lambda Theorem

    lemmalem:dynkin-pi-lambda-2026aAnalysisProbability
    Let XX be a set. A family P\mathcal{P} of subsets of XX is a 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 λ\lambda-system if: (1) X∈LX\in\mathcal{L}; (2…

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    Authors Claude-agent-v1, Aaron · Created

  • Product Sigma-Algebra

    definitiondef:product-sigma-algebra-2026aAnalysisProbability
    Let (X,F)(X,\mathcal{F}) and (Y,G)(Y,\mathcal{G}) be measurable spaces. A measurable rectangle is a subset of the 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 product σ\sigma-algebra F⊗G\mathcal{F}\otimes\mathcal{G} on…

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    Authors Claude-agent-v1, Aaron · Created

  • Linearity and Monotonicity of the Lebesgue Integral

    theoremthm:linearity-monotonicity-integral-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a measure space. 1. (Nonnegative case.) Let f,g:X→[0,∞]f,g:X\to[0,\infty] be 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 con…

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    Authors Claude-agent-v1, Aaron · Created

  • Dominated Convergence Theorem

    theoremthm:dominated-convergence-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a measure space, and let (fm)m∈N(f_m)_{m\in\mathbb{N}} be a sequence of 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 converges to f(x)f(x), for a function f:X→Rf:X\to\mathbb{R}. Suppose there is an integrable…

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    Authors Claude-agent-v1, Aaron · Created

  • Let (X,F,μ)(X,\mathcal{F},\mu) be a measure space and let (fm)m∈N(f_m)_{m\in\mathbb{N}} be a sequence of 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 infimum and supremum are taken i…

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    Authors Claude-agent-v1, Aaron · Created

  • Monotone Convergence Theorem

    theoremthm:monotone-convergence-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a measure space and let (fm)m∈N(f_m)_{m\in\mathbb{N}} be a sequence of 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…

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    Authors Claude-agent-v1, Aaron · Created

  • Integrable Function and the Lebesgue Integral

    definitiondef:lebesgue-integral-integrable-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a measure space and let f:X→Rf:X\to\mathbb{R} be measurable. Define the positive part f+f^{+} and 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…

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    Authors Claude-agent-v1, Aaron · Created

  • Lebesgue Integral of a Nonnegative Measurable Function

    definitiondef:lebesgue-integral-nonnegative-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a measure space. A function f:X→[0,∞]f:X\to[0,\infty] (values in the extended half-line of Measure, Measure Space, and Probability Measure) is called 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 ag…

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    Authors Claude-agent-v1, Aaron · Created

  • Simple Function and Its Integral

    definitiondef:simple-function-integral-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a measure space. For A⊆XA\subseteq X, the 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 simple function on (X,F)(X,\mathcal{F}) is a measurable function…

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    Authors Claude-agent-v1, Aaron · Created

  • Let (X,F)(X,\mathcal{F}) and (Y,G)(Y,\mathcal{G}) be measurable spaces. A function f:X→Yf:X\to Y is 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 measurable if it…

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    Authors Claude-agent-v1, Aaron · Created

  • Let λ∗\lambda^{*} be the Lebesgue outer measure on the real line. Then: 1. λ∗\lambda^{*} is an outer measure on R\mathbb{R}; 2. every Borel set is Carathéodory measurable with respect to λ∗\lambda^{*}; 3. consequently, by Caratheodory Extension Theorem, the restriction…

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    Authors Claude-agent-v1, Aaron · Created

  • Lebesgue Outer Measure on the Real Line

    definitiondef:lebesgue-outer-measure-real-line-2026aAnalysisProbability
    For a subset AA of the real line R\mathbb{R}, the 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 infimum is taken over all sequences of open intervals (am,bm)(a_m,b_m)…

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    Authors Claude-agent-v1, Aaron · Created

  • Caratheodory Extension Theorem

    theoremthm:caratheodory-extension-2026aAnalysisProbability
    Let XX be a set and let μ∗\mu^{*} be an 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: 1. M\mathcal{M} is a σ\sigma-algebra on XX; 2. the restricti…

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    Authors Claude-agent-v1, Aaron · Created

  • Let XX be a set. An 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 Measure, Measure Space, and Probability Measure) such that: 1. μ∗(∅)=0\mu^{*}(\varnothing)=0; 2. (monotonicity) if…

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    Authors Claude-agent-v1, Aaron · Created

  • Let (X,F)(X,\mathcal{F}) be a 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…

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    Authors Claude-agent-v1, Aaron · Created

  • Borel Sigma-Algebra on the Real Line

    definitiondef:borel-sigma-algebra-real-line-2026aAnalysisProbability
    Identify the real line R\mathbb{R} with the Euclidean space R1\mathbb{R}^1. The Borel σ\sigma-algebra on R\mathbb{R}, denoted B(R)\mathcal{B}(\mathbb{R}), is the σ\sigma-algebra generated by the family of all open subsets of R\mathbb{R}. Its members are called Borel sets. In…

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    Authors Claude-agent-v1, Aaron · Created

  • Generated Sigma-Algebra

    definitiondef:generated-sigma-algebra-2026aAnalysisProbability
    Let XX be a set and let C\mathcal{C} be a family of subsets of XX. The intersection of any nonempty collection of 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…

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    Authors Claude-agent-v1, Aaron · Created

  • Sigma-Algebra and Measurable Space

    definitiondef:sigma-algebra-measurable-space-2026aAnalysisProbability
    Let XX be a set. A σ\sigma-algebra on XX is a family F\mathcal{F} of subsets of XX with the following three properties. 1. X∈FX\in\mathcal{F}. 2. If A∈FA\in\mathcal{F}, then the complement X∖AX\setminus A belongs to F\mathcal{F}. 3. For every sequence (Am)m∈N(A_m)_{m\in\mathbb{N}}…

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    Authors Claude-agent-v1, Aaron · Created

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