Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
- A linear functional on the continuous functions with compact support on a metric space is positive if it takes nonnegative values at nonnegative functions.
- Standing notation for the duality between continuous functions with compact support and finite signed Borel measures on a metric space: the dual space of C_c(X) with its dual norm, the space M(X) with the total variation norm, and the pointwise order of real functions.
Nonnegative Combinations of Finite Measures and Their Integrals
lemmalem:nonnegative-combination-finite-measures-2026aAnalysisA nonnegative linear combination of two finite measures is a finite measure, and integrals against it are the same combination of the integrals against the two measures, both for nonnegative measurable functions and for functions integrable against both.Finite Signed Radon Measure on a Metric Space
definitiondef:signed-radon-measure-metric-2026aAnalysisA finite signed Borel measure on a metric space is a finite signed Radon measure when its total variation is a Radon measure.Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures
lemmalem:integral-signed-measure-properties-2026aAnalysisProbabilityThe integral against a finite signed measure is linear, bounded by the integral of the absolute value against the total variation, defined for bounded measurable functions with bound C times the total variation norm, computable from any splitting of the measure as a difference of…Integrable Functions and the Integral with Respect to a Finite Signed Measure
definitiondef:integral-signed-measure-2026aAnalysisProbabilityA real function is integrable with respect to a finite signed measure if it is integrable with respect to its total variation, and its integral is the difference of its integrals against the two parts of the Jordan decomposition.Properties of the Total Variation: a Finite Measure, the Jordan Decomposition, Linear Combinations, the Total Variation Norm, and Integrability
lemmalem:total-variation-properties-2026aAnalysisProbabilityThe total variation of a finite signed measure is a finite measure equal to the sum of the Jordan parts and dominating the measure; it is subadditive and homogeneous under setwise linear combinations, so the total variation norm makes the finite signed measures a real normed spac…Total Variation and Total Variation Norm of a Finite Signed Measure
definitiondef:total-variation-measure-2026aAnalysisProbabilityDefines the total variation of a finite signed measure, the set function giving on each measurable set A the largest value of ν(B) − ν(A∖B) over measurable B ⊆ A, and the total variation norm, its value on the whole space.Existence of the Total Variation of a Finite Signed Measure
lemmalem:total-variation-existence-2026aAnalysisProbabilityFor a finite signed measure and a measurable set A, the numbers ν(B) − ν(A∖B) over the measurable subsets B of A have a greatest element, attained at the part of A in any Hahn set and equal to ν⁺(A) + ν⁻(A).Radon Measures on Locally Compact Metric Spaces: Standing Notation
settingset:radon-measures-notation-2026aAnalysisProbabilityCombines the notation for locally compact metric spaces and compactly supported functions with that for measures and the Lebesgue integral, and fixes the terms Borel measure and finite signed Borel measure, the setwise operations on finite signed measures, and the Jordan decompos…Radon Measures and Finite Signed Borel Measures Are Determined by Their Integrals of Compactly Supported Continuous Functions
lemmalem:radon-determined-by-cc-2026aAnalysisProbabilityOn a locally compact metric space every continuous function with compact support is integrable with respect to every Radon measure and every finite signed Radon measure; two Radon measures with the same integrals of these functions are equal, and on a separable such space so are…Finite Borel Measures on a Locally Compact Separable Metric Space Are Radon
theoremthm:finite-borel-radon-locally-compact-2026aAnalysisProbabilityOn a locally compact separable metric space, a finite Borel measure is inner regular by compact sets on all Borel sets and is a Radon measure, and every finite signed Borel measure is a finite signed Radon measure.Closed–Open Regularity of a Finite Borel Measure on a Metric Space
lemmalem:regularity-finite-borel-metric-2026aAnalysisProbabilityFor a finite Borel measure on a metric space, every Borel set lies between a closed set and an open set whose difference has arbitrarily small measure.- A Borel measure on a metric space is a Radon measure if it is finite on compact sets, outer regular by open sets and inner regular on open sets by compact sets.
The Space of Continuous Functions with Compact Support on a Locally Compact Separable Metric Space is Separable
lemmalem:compactly-supported-continuous-separable-2026aAnalysisTopologyFor a locally compact separable metric space X, the real normed space of continuous compactly supported functions on X with the uniform norm is separable.Exhaustion of a Locally Compact Separable Metric Space by Compact Sets, with Cut-Off Functions
lemmalem:exhaustion-locally-compact-separable-2026aAnalysisTopologyA locally compact separable metric space is the union of an increasing sequence of compact sets, each inside the interior of the next, such that every compact set lies inside one of them; and there are continuous compactly supported cut-off functions equal to 1 on each set and su…Thickenings, Neighbourhoods with Compact Closure, Urysohn Functions and Partitions of Unity in a Locally Compact Metric Space
lemmalem:urysohn-locally-compact-metric-2026aAnalysisTopologyIn a locally compact metric space, a compact set inside an open set has a compact closed thickening and an open neighbourhood with compact closure inside the open set, and a continuous compactly supported function with values in [0,1] that is 1 on the compact set and supported in…Compact Subsets of a Metric Space: Covers by Open Sets, Finite Unions and Increasing Covers; Sublevel Sets of the Distance and Closed Balls
lemmalem:compact-subsets-metric-covers-unions-2026aAnalysisTopologyA subset of a metric space is compact exactly when every cover by open subsets of the space has a finite subcover; finite unions of compact sets are compact, a compact set covered by an increasing sequence of open sets lies in one of them, and the sets where the distance to a set…The Continuous Functions with Compact Support Form a Real Normed Space under the Uniform Norm
lemmalem:compactly-supported-continuous-normed-2026aAnalysisTopologyC_c(X) with pointwise operations is a real vector space and the uniform norm makes it a real normed space; the uniform norm bounds every value and is the least nonnegative such bound, and convergence in it is uniform convergence.The Space of Continuous Functions with Compact Support and the Uniform Norm
definitiondef:compactly-supported-continuous-2026aAnalysisTopologyC_c(X) is the set of continuous real functions with compact support on a metric space X, with pointwise sum and scalar multiplication, and the uniform norm of f is the largest of 0 and the values |f(x)|.