Measures are monotone

Depends on

• Measures on sigma-algebras

Used by

• One-dimensional W^1,p functions have unique absolutely continuous representatives Corollary
• Poincare-Wirtinger on bounded convex domains by the direct pairwise argument Corollary
• Polar integration may discard the cut locus Corollary
• The obstacle reaction is supported on the contact set under measure regularity Corollary
• A null set can fail to be the discontinuity set of any function Counterexample
• Neumann Poisson data require a flux compatibility equation Counterexample
• Poincare-Wirtinger fails on disconnected bounded domains Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• The distribution function of a Borel measure on ℝ, normalized at 0 Definition
• The Dirichlet function satisfies Lusin's conclusion without being continuous anywhere Example
• A finite-measure measurable set in ℝⁿ is approximable in measure by a finite union of boxes Lemma
• A measurable set of positive finite measure occupies more than any prescribed proportion of some dyadic cube Lemma
• A separated characteristic disk has a minimal nonidentity simple cycle Lemma
• An area-minimal three-sector homoclinic cycle has identity inward holonomy Lemma
• Atoms have uniformly bounded Hᵖ quasi-norm and uniformly bounded test pairings Lemma
• Ball and cube maximal functions are pointwise comparable Lemma
• Basic identities for a probability measure Lemma
• Bounded-overlap ball chains in a bounded John domain Lemma
• Dyadic annulus far-field estimates for the maximal function Lemma
• Euclidean balls have positive finite Lebesgue measure Lemma
• L2-normalised H1 atoms have uniformly bounded H1 norm Lemma
• Maximal dyadic subcubes of a cube at a height Lemma
• Mean-zero Poincare estimate on bounded connected extension domains below the dimension Lemma
• Near and far bounds for a Riesz potential Lemma
• No-return sets have disjoint null preimage towers Lemma
• Radially decreasing kernels are dominated by the maximal function Lemma
• Standard Hölder kernels satisfy the Hörmander condition Lemma
• The truncated Riesz kernel is bounded on Lᵖ of a bounded set Lemma
• Whitney-type ball cover with disjoint small balls and bounded overlap Lemma
• Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure Proposition
• Measure of a set difference when the smaller set has finite measure Proposition
• Null sets are closed under countable unions and, in a complete space, under arbitrary subsets Proposition
• Sets whose symmetric difference is null have the same measure Proposition
• Almost uniform convergence implies almost-everywhere convergence and convergence in measure Theorem
• Assuming countable choice, the Lebesgue measure of a measurable set is the supremum of the measures of its compact subsets Theorem
• Assuming countable choice, the semifinite part is a semifinite measure and equals the original measure exactly when it is semifinite Theorem
• Egorov's theorem Theorem
• Finite and countable subadditivity of measures Theorem
• If a Lebesgue measurable subset of ℝⁿ has positive measure, its difference set contains an open ball about the origin Theorem
• Lebesgue outer measure is at most Jordan outer content, and a bounded Jordan measurable set is Lebesgue measurable with Lebesgue measure equal to its Jordan content Theorem

…and 7 more results.

Dependency tree · two levels

4 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 2.57 (standard reference, not scraped)