Continuity from below for measures

Depends on

• Measures on sigma-algebras
• Measure of a set difference when the smaller set has finite measure
• Series in the nonnegative extended real line
• Every subset of $\overline{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\overline{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$

Used by

• Density inversion from an integrable characteristic function Corollary
• A nontight sequence with no probability law subsequence limit Counterexample
• An open dense set of measure less than 1 is the monotone L¹-limit of Riemann integrable indicators, but its indicator is not Riemann integrable Example
• Exponential martingale Brownian tail bound Example
• Green kernel of a biased integer walk Example
• The arctangent distribution function generates a Borel probability measure Example
• A finite-measure measurable set in ℝⁿ has a compact core and a bounded open neighbourhood of arbitrarily small excess Lemma
• A finite-measure measurable set in ℝⁿ is approximable in measure by a finite union of boxes Lemma
• A universal Martin-Löf test exists Lemma
• Approximation in symmetric difference by a generating algebra Lemma
• Assuming countable choice, an infinite-measure set in a semifinite measure space has arbitrarily large finite-measure subsets Lemma
• Basic identities for a probability measure Lemma
• Bounded centered convergent series have summable variances Lemma
• Chacon partial maps extend to an invertible map mod null sets Lemma
• Dyadic coding supplies coin measure and its completed Lebesgue transfer Lemma
• Exact L2 Fourier multiplier norm Lemma
• Finite measures agreeing on a generating pi-system and on the whole space are equal Lemma
• Finite-measure sets are approximable in measure by sets from a countable generating algebra Lemma
• Kolmogorov block polynomial with large partial sums Lemma
• Lipschitz curves and dominated interval vector measures Lemma
• Nondentability produces a vector measure without density Lemma
• Probability algebras, arbitrary joins and the countable chain condition Lemma
• Product rectangle kernels are dense in product L two Lemma
• Real cdf and bounded continuous definitions agree Lemma
• Simultaneous rational conditional distribution function versions Lemma
• Tail comparisons under independent-copy symmetrization Lemma
• The sharp interval bound for Cantor measure Lemma
• Assuming countable choice, the Carathéodory extension dominates every other extension and agrees with it on finite-measure sets Proposition
• Borel cores of sigma-finite Hausdorff measurable sets Proposition
• For sigma-finite measures, the section-measure functions are measurable Proposition
• The Cantor measure is a singular atomless probability measure concentrated on the Cantor set Proposition
• Assuming countable choice, Borel probability measures on Polish spaces are inner regular Theorem
• Assuming countable choice, finite-on-compacts Borel measures on ℝ correspond to nondecreasing right-continuous functions modulo constants Theorem
• Assuming countable choice, Lusin's theorem on finite-measure subsets of Rⁿ Theorem
• Assuming countable choice, regular outer measures are continuous from below on all subsets Theorem
• Assuming countable choice, the Lebesgue measure of a measurable set is the supremum of the measures of its compact subsets Theorem
• Brownian time inversion Theorem
• Continuity from above when one set has finite measure Theorem
• Continuity from below, and from above when one set has finite signed measure Theorem
• Convergence in probability and almost surely agree for independent series Theorem

…and 19 more results.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• S. Axler, Measure, Integration & Real Analysis, Theorem 2.59 (standard reference, not scraped)