Continuity from above when one set has finite measure

Depends on

• Continuity from below for measures
• Measure of a set difference when the smaller set has finite measure
• 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

• A nondecreasing function that is not right-continuous can fail countable additivity Counterexample
• Counting-measure tails decrease to the empty set while every term has infinite measure Counterexample
• Borel master codes for null and meagre sets Definition
• Birth–death recurrence through scale products Example
• Borel's exceptional set can be uncountable and null Example
• The fat Cantor set has positive length and dimension one Example
• The Smith-Volterra-Cantor set has Lebesgue measure exactly 1/2 Example
• FALSE: continuity from above needs no finiteness hypothesis False statement
• FALSE: every nondecreasing function defines a Lebesgue-Stieltjes measure on Borel sets False statement
• Basic identities for a probability measure Lemma
• Dyadic coding supplies coin measure and its completed Lebesgue transfer Lemma
• Levy prokhorov distance is a metric Lemma
• Maximum principle for a compact logarithmic potential Lemma
• Real cdf and bounded continuous definitions agree Lemma
• The countable-product cylinder premeasure is countably additive Lemma
• Transfer of null and meagre invariants between Cantor space and the line Lemma
• Uniform open hulls for Borel sections of small measure Lemma
• For sigma-finite measures, the section-measure functions are measurable Proposition
• Assuming countable choice, finite-on-compacts Borel measures on ℝ correspond to nondecreasing right-continuous functions modulo constants Theorem
• Cauchy sequences in measure converge in 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
• Converging together lemma Theorem
• Egorov's theorem Theorem
• Equivalent criteria for recurrence and transience Theorem
• Finite-total-variation signed measures are complete Theorem
• Interval formulas and atoms for a Lebesgue-Stieltjes measure Theorem
• Kac integral formula for excursions Theorem
• Kac return-time formula without invertibility Theorem
• Kolmogorov convergence criterion Theorem
• Lebesgue-Stieltjes measures on ℝ are outer regular and inner regular by compact sets Theorem
• Levy prokhorov metric metrizes weak convergence Theorem
• On a finite measure space, almost-everywhere convergence implies convergence in measure Theorem
• Portmanteau theorem Theorem
• Probability laws correspond to distribution functions Theorem
• Riesz's subsequence theorem for convergence in measure Theorem
• The limsup of the measures is at most the measure of the set limsup under a finite-union bound Theorem

Dependency tree · two levels

14 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 2.60 (standard reference, not scraped)