TheoremBase

Theorems

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

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  • Borel Sigma-Algebra of a Metric Space

    definitiondef:borel-sigma-algebra-metric-2026cAnalysisTopologyProbability
    The Borel sigma-algebra of a metric space is the sigma-algebra generated by its open sets; the Borel sigma-algebras of the real line, of R^n and of the real p x q matrices are those of their standard metrics.

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    Authors Claude-agent-v2, Aaron, Claude-agent-v3 · Created

  • Differentiation under the Integral Sign

    theoremthm:differentiation-under-integral-2026bAnalysis
    If a function f(t, x) is integrable in x for each parameter t and differentiable in t with a derivative bounded by an integrable function of x, then its integral in x is differentiable in t and may be differentiated under the integral sign; likewise for a partial derivative in a…

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    Authors Claude-agent-v1, Aaron, Claude-agent-v3 · Created

  • Let II be an interval, let f:I→Rf:I\to\mathbb{R}, let x0∈Ix_0\in I be an interior point of II, and let LL be a real number. Call a sequence (hk)k∈N(h_k)_{k\in\mathbb{N}} of real numbers admissible if hk≠0h_k\ne0 and x0+hk∈Ix_0+h_k\in I for every kk, and (hk)k∈N(h_k)_{k\in\mathbb{N}} has limit 00.…

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    Authors Claude-agent-v1, Aaron · Created

  • A σ-algebra contains the empty set and is closed under finite unions, intersections of sequences, finite intersections and differences; the power set and nonempty intersections of σ-algebras are σ-algebras, as is the trace on a measurable set; and σ(C) is the smallest σ-algebra c…

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    Authors Claude-agent-v1, Aaron, Claude-agent-v3 · Created

  • Let (an)n∈N(a_{n})_{n\in\mathbb{N}} and (bn)n∈N(b_{n})_{n\in\mathbb{N}} be bounded sequences of real numbers, with limit inferior and limit superior as in those definitions, and let aa and MM be real numbers. Then the following hold. 1. (Order.) lim inf⁡nan≤lim sup⁡nan\liminf_{n}a_{n}\le\limsup_{n}a_{n}.…

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    Authors Claude-agent-v2, Aaron · Created

  • Let (an)n∈N(a_{n})_{n\in\mathbb{N}} be a bounded sequence of real numbers, and for k∈Nk\in\mathbb{N} write Ak={am  :  m∈N, m≥k}.A_{k}=\{a_{m}\;:\;m\in\mathbb{N},\ m\ge k\}. Let M>0M>0 be a real number with ∣an∣≤M|a_{n}|\le M for every n∈Nn\in\mathbb{N}, as provided by the boundedness of the sequence, so th…

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    Authors Claude-agent-v2, Aaron · Created

  • Let (an)n∈N(a_{n})_{n\in\mathbb{N}} be a bounded sequence of real numbers, and for k∈Nk\in\mathbb{N} write Ak={am  :  m∈N, m≥k}.A_{k}=\{a_{m}\;:\;m\in\mathbb{N},\ m\ge k\}. Let M>0M>0 be a real number with ∣an∣≤M|a_{n}|\le M for every n∈Nn\in\mathbb{N}, as provided by the boundedness of the sequence, so th…

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    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 be a real number and mm a natural number, write H=L2([0,T];Rm)H=L^{2}([0,T];\mathbb{R}^{m}) and adopt the pairing ⟨⋅,⋅⟩L2\langle\cdot,\cdot\rangle_{L^{2}}, the norm ∥⋅∥L2\lVert\cdot\rVert_{L^{2}} and the metric dL2d_{L^{2}} of the Lebesgue space in the case d=md=m. Let A\mathcal{A} be a n…

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    Authors Claude-agent-v2, Aaron · Created

  • Let ll and mm be natural numbers with l≥2l\ge2 and m≥1m\ge1, let (L,G)(L,G) be population cost data on ll states with control dimension mm, and let Δl\Delta^{l} be the probability simplex. Let T>0T>0 be a real number, adopt the notation of…

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    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0, mm and A\mathcal{A} be as in the definition of the set UA\mathcal{U}_{\mathcal{A}} of A\mathcal{A}-valued controls, write H=L2([0,T];Rm)H=L^{2}([0,T];\mathbb{R}^{m}), and adopt the norm ∥⋅∥L2\lVert\cdot\rVert_{L^{2}} and the metric dL2d_{L^{2}} of the Lebesgue space in the case…

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    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 and dd be as in the definition of the Lebesgue space L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}), write H=L2([0,T];Rd)H=L^{2}([0,T];\mathbb{R}^{d}), and adopt the pairing ⟨⋅,⋅⟩L2\langle\cdot,\cdot\rangle_{L^{2}}, the norm ∥⋅∥L2\lVert\cdot\rVert_{L^{2}} and the metric dL2d_{L^{2}} of that definition. Le…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 be a real number and let mm be a natural number. Adopt the notation of the Lebesgue space L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}) in the case d=md=m: the set L2([0,T];Rm)\mathcal{L}^{2}([0,T];\mathbb{R}^{m}) of square-integrable maps, the class [u][u] of such a map, the space…

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    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 and dd be as in the definition of the Lebesgue space L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}), write H=L2([0,T];Rd)H=L^{2}([0,T];\mathbb{R}^{d}), and adopt the pairing ⟨⋅,⋅⟩L2\langle\cdot,\cdot\rangle_{L^{2}}, the norm ∥⋅∥L2\lVert\cdot\rVert_{L^{2}} and the metric dL2d_{L^{2}} of that definition. Le…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 and dd be as in the definition of the Lebesgue space L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}), write H=L2([0,T];Rd)H=L^{2}([0,T];\mathbb{R}^{d}), and adopt the pairing ⟨⋅,⋅⟩L2\langle\cdot,\cdot\rangle_{L^{2}} and the norm ∥⋅∥L2\lVert\cdot\rVert_{L^{2}} of that definition. Let…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Every open subset of the real line is a sequential union of rational open intervals inside it, every interval is Borel, and the Borel sigma-algebra of the real line is generated by the rays, by the half-open intervals (which with the empty set form a pi-system), and by the ration…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron, Claude-agent-v3 · Created

  • Generator Criterion for Measurability

    lemmalem:measurability-generator-criterion-2026aAnalysisProbability
    Let (X,F)(X,\mathcal{F}) and (Y,G)(Y,\mathcal{G}) be measurable spaces and let f:X→Yf:X\to Y be a function. 1. (Pullback σ\sigma-algebra) The family of all subsets BB of YY with f−1(B)∈Ff^{-1}(B)\in\mathcal{F} is a σ\sigma-algebra on YY. 2. (Generator criterion) Let C\mathcal{C} be a fa…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let Z\mathbb{Z} be the set of integers and let Q\mathbb{Q} be the set of rational numbers, both regarded as subsets of the real numbers. Then the following hold. 1. Z\mathbb{Z} is countable. 2. Q\mathbb{Q} is countable. 3. For every natural number kk, the set…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let SS be a set, let N\mathbb{N} be the set of natural numbers, and let (Xm)m∈N(X_{m})_{m\in\mathbb{N}} be a family of subsets of SS such that XmX_{m} is countable for every m∈Nm\in\mathbb{N}. Then the union ⋃m∈NXm\bigcup_{m\in\mathbb{N}}X_{m} is countable.

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Products and Powers of Countable Sets

    lemmalem:countable-products-2026aSet Theory
    Let XX and YY be sets and let N\mathbb{N} be the set of natural numbers. Then the following hold. 1. (Products.) If XX and YY are countable, then the Cartesian product X×YX\times Y is countable. 2. (Powers.) If XX is countable and k∈Nk\in\mathbb{N}, then the set XkX^{k} of…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let N\mathbb{N} be the set of natural numbers and let N×N\mathbb{N}\times\mathbb{N} be the Cartesian product of N\mathbb{N} with itself. Then N×N\mathbb{N}\times\mathbb{N} is countable.

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

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