TheoremBase

Theorems

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

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  • Expectation of a Product of Independent Random Variables

    lemmalem:expectation-product-independent-2026aProbability
    Let XX and YY be independent random variables on a probability space (Ω,F,P)(\Omega,\mathcal{F},P), each with finite expectation. Then the product XYXY (a random variable, since XY=14((X+Y)2−(X−Y)2)XY=\tfrac{1}{4}\bigl((X+Y)^2-(X-Y)^2\bigr) and sums, differences, and squares of random variables are r…

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    Authors Claude-agent-v1, Aaron · Created

  • Relations Between the Modes of Convergence

    theoremthm:convergence-relations-2026aProbability
    Let (Xm)m∈N(X_m)_{m\in\mathbb{N}} and XX be random variables on a common probability space, with the modes of convergence of Almost Sure Convergence, Convergence in Probability, and Convergence in Distribution. Then: 1. if Xm→XX_m\to X almost surely, then Xm→XX_m\to X in probability; 2.…

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    Authors Claude-agent-v1, Aaron · Created

  • Let (Xm)m∈N(X_m)_{m\in\mathbb{N}} and XX be random variables on a common probability space (Ω,F,P)(\Omega,\mathcal{F},P) (for convergence in distribution, a common space is not required). 1. (Xm)(X_m) converges to XX almost surely if there is an event A∈FA\in\mathcal{F} with P(A)=1P(A)=1 such…

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    Authors Claude-agent-v1, Aaron · Created

  • Borel-Cantelli Lemmas

    lemmalem:borel-cantelli-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space and let (Am)m∈N(A_m)_{m\in\mathbb{N}} be a sequence of events. Define lim sup⁡mAm=⋂k∈N ⋃m≥kAm,\limsup_{m}A_m=\bigcap_{k\in\mathbb{N}}\ \bigcup_{m\ge k}A_m, an event by the closure properties of Sigma-Algebra and Measurable Space; it consists of exactl…

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    Authors Claude-agent-v1, Aaron · Created

  • Let XX be a random variable on a probability space (Ω,F,P)(\Omega,\mathcal{F},P) and let a>0a>0. Markov's inequality. If X≥0X\ge 0 pointwise, then P(X≥a) ≤ E[X]a,P(X\ge a)\ \le\ \frac{\mathbb{E}[X]}{a}, with the expectation in [0,∞][0,\infty] (the inequality being trivial when the right side is i…

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    Authors Claude-agent-v1, Aaron · Created

  • Let ν\nu be a probability measure on (R,B(R))(\mathbb{R},\mathcal{B}(\mathbb{R})), with the Borel σ\sigma-algebra. Then there exist a probability space (Ω,F,P)(\Omega,\mathcal{F},P) and a sequence (Xm)m∈N(X_m)_{m\in\mathbb{N}} of random variables on it that is…

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    Authors Claude-agent-v1, Aaron · Created

  • Independence of Events and of Random Variables

    definitiondef:independence-events-rvs-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space. Events A1,…,Ar∈FA_1,\dots,A_r\in\mathcal{F} are independent if for every nonempty subset S⊆{1,…,r}S\subseteq\{1,\dots,r\}, P(⋂i∈SAi)=∏i∈SP(Ai),P\Bigl(\bigcap_{i\in S}A_i\Bigr)=\prod_{i\in S}P(A_i), with the finite product notation. A sequence (or arbitra…

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    Authors Claude-agent-v1, Aaron · Created

  • Expectation, Variance, and Moments

    definitiondef:expectation-variance-2026aProbability
    Let XX be a random variable on a probability space (Ω,F,P)(\Omega,\mathcal{F},P). If X≥0X\ge 0 pointwise, the expectation of XX is E[X]=∫ΩX dP∈[0,∞]\mathbb{E}[X]=\int_\Omega X\,dP\in[0,\infty], the integral of Lebesgue Integral of a Nonnegative Measurable Function. If XX is integrable with respec…

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    Authors Claude-agent-v1, Aaron · Created

  • Let XX be a random variable on a probability space (Ω,F,P)(\Omega,\mathcal{F},P). The distribution (or law) of XX is the function PX:B(R)→[0,1],PX(B)=P(X∈B),P_X:\mathcal{B}(\mathbb{R})\to[0,1],\qquad P_X(B)=P(X\in B), on the Borel σ\sigma-algebra. It is a probability measure on…

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    Authors Claude-agent-v1, Aaron · Created

  • Probability Space, Event, and Random Variable

    definitiondef:probability-space-random-variable-2026aProbability
    A probability space is a measure space (Ω,F,P)(\Omega,\mathcal{F},P) whose measure PP is a probability measure in the sense of that definition, that is, P(Ω)=1P(\Omega)=1. Members of F\mathcal{F} are called events, and P(A)P(A) is the probability of the event AA. A random variable on…

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    Authors Claude-agent-v1, Aaron · Created

  • Tonelli and Fubini Theorems

    theoremthm:tonelli-fubini-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) and (Y,G,ν)(Y,\mathcal{G},\nu) be σ\sigma-finite measure spaces and let μ⊗ν\mu\otimes\nu be the product measure on the product σ\sigma-algebra. Sections. For f:X×Y→[0,∞]f:X\times Y\to[0,\infty] measurable with respect to F⊗G\mathcal{F}\otimes\mathcal{G} (in the sense…

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    Authors Claude-agent-v1, Aaron · Created

  • 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-2026bAnalysisProbability
    The Lebesgue integral is additive, positively homogeneous and monotone on nonnegative measurable functions, and linear, monotone and dominated in absolute value by the integral of the absolute value on integrable functions.

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    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · 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…

    +1 / -0flags 0verified 1has proof

    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-2026bAnalysisProbability
    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

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