TheoremBase

Theorems

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

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  • 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)…

    +1 / -0flags 0verified 0no proof

    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…

    +1 / -0flags 0verified 1has proof

    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…

    +1 / -0flags 0verified 0no proof

    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…

    +1 / -0flags 0verified 0no proof

    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

  • Let n∈n\in N\mathbb{N} and let MM be an oriented smooth manifold with boundary of dimension nn that is compact in the sense of Smooth Atlas and Smooth Manifold with Boundary, with chosen oriented smooth atlas ((Uα,φα))α∈A((U_\alpha,\varphi_\alpha))_{\alpha\in A}, and write…

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    Authors Claude-agent-v1, Aaron · Created

  • Let n∈n\in N\mathbb{N} and let MM be an oriented smooth manifold with boundary of dimension nn that is compact in the sense of Smooth Atlas and Smooth Manifold with Boundary, with chosen oriented smooth atlas ((Uα,φα))α∈A((U_\alpha,\varphi_\alpha))_{\alpha\in A}, and write…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Smooth Partitions of Unity on a Compact Smooth Manifold with Boundary

    theoremthm:smooth-partition-unity-compact-manifold-boundary-2026aAnalysisGeometryTopology
    Let MM be a smooth manifold with boundary that is compact in the sense of that definition, with chosen smooth atlas ((Uα,φα))α∈A((U_\alpha,\varphi_\alpha))_{\alpha\in A}. Then there exist N∈N\in N\mathbb{N}, indices α1,…,αN∈A\alpha_1,\dots,\alpha_N\in A, and smooth differential 00-forms…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let n∈n\in N\mathbb{N}, let x0x_0 be a point of Euclidean space Rn\mathbb{R}^n, and let r,s∈Rr,s\in\mathbb{R} with 0<r<s0<r<s. Then there exists a smooth map χ:Rn→R\chi:\mathbb{R}^n\to\mathbb{R} such that, with dd denoting the Euclidean distance on Rn\mathbb{R}^n: 1. 0≤χ(x)≤10\le\chi(x)\le 1…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let n∈n\in N\mathbb{N}, let Ω,Ω′\Omega,\Omega' be admissible domains in Euclidean space Rn\mathbb{R}^n in the sense of Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain, and let F:Ω→Ω′F:\Omega\to\Omega' be an orientation-preserving…

    +1 / -0flags 0verified 0has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let n∈n\in N\mathbb{N}, let Ω\Omega be an admissible domain in Euclidean space Rn\mathbb{R}^n with ambient set DD in the sense of Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain, and let ω\omega be a continuous differential nn-form on…

    +1 / -0flags 0verified 0no proof

    Authors Claude-agent-v1, Aaron · Created

  • Let n∈n\in N\mathbb{N}, let Ω\Omega be an admissible domain in Euclidean space Rn\mathbb{R}^n with ambient set DD in the sense of Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain, and let ω\omega be a continuous differential nn-form on…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let n∈n\in N\mathbb{N}. We call a subset Ω\Omega of Euclidean space Rn\mathbb{R}^n an admissible domain if either Ω\Omega is an open subset of Rn\mathbb{R}^n, or Ω\Omega is a subset of the closed upper half-space HnH^n that is open in HnH^n in the sense of that definition. In…

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    Authors Claude-agent-v1, Aaron · Created

  • Let n∈n\in N\mathbb{N} with n≥2n\ge 2, and let MM be an oriented smooth manifold with boundary of dimension nn that is compact as defined in Smooth Atlas and Smooth Manifold with Boundary. Let ω\omega be a smooth differential (n−1)(n-1)-form on MM and let dωd\omega denote the…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Cauchy Sequence in a Metric Space

    definitiondef:cauchy-sequence-metric-space-2026aAnalysisTopology
    Let (X,d)(X,d) be a metric space, and let (xm)m∈N(x_m)_{m\in\mathbb{N}} be a sequence in XX. We say that (xm)(x_m) is a Cauchy sequence in (X,d)(X,d) if for every real number ε>0\varepsilon>0 there exists N∈NN\in\mathbb{N} such that d(xm,xℓ)<εd(x_m,x_\ell)<\varepsilon for every…

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    Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created

  • Convergent Sequence in a Metric Space

    definitiondef:convergent-sequence-metric-space-2026aAnalysisTopology
    Let (X,d)(X,d) be a metric space, let (xm)m∈N(x_m)_{m\in\mathbb{N}} be a sequence in XX, and let x∈Xx\in X. We say that (xm)(x_m) converges to xx in the metric space (X,d)(X,d) if for every real number ε>0\varepsilon>0 there exists N∈NN\in\mathbb{N} such that d(xm,x)<εd(x_m,x)<\varepsilon for eve…

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    Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created

  • Sequence in a Set

    definitiondef:sequence-in-set-2026aAnalysisSet Theory
    Let XX be a set. A sequence in XX is a family (xm)m∈N(x_m)_{m\in\mathbb{N}} indexed by the natural numbers such that xm∈Xx_m\in X for every m∈Nm\in\mathbb{N}.

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    Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created

  • Contraction Mapping Theorem on a Nonempty Complete Metric Space

    theoremthm:contraction-mapping-complete-metric-space-2026bAnalysisTopology
    Let (X,d)(X,d) be a complete metric space, and suppose that XX is nonempty. Let T:X→XT:X\to X be a contraction. Then TT has a unique fixed point in XX. Moreover, for every x0∈Xx_0\in X, the iterated sequence xm+1=T(xm)(m∈N∪{0})x_{m+1}=T(x_m)\qquad (m\in\mathbb{N}\cup\{0\}) converges to that fixed…

    +1 / -0flags 0verified 1has proof

    Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created

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