Finite and countable subadditivity of measures

Depends on

• Measures on sigma-algebras
• Measures are monotone
• Series in the nonnegative extended real line
• The well-ordering principle

Used by

• Brownian paths are locally Holder below one half Corollary
• Dominated convergence is a Vitali corollary Corollary
• Polar integration may discard the cut locus Corollary
• Submartingale doob decomposition has increasing compensator Corollary
• The Solovay model has no Vitali or Bernstein set Corollary
• Topological recurrence on second-countable spaces Corollary
• A nonintegrable observable with divergent ergodic averages Counterexample
• A null set can fail to be the discontinuity set of any function Counterexample
• Assuming Choice, a proper subgroup of (ℝ,+) can be nonmeasurable Counterexample
• Cantor function has singular distributional derivative Counterexample
• Hilbert transform does not map L-infinity to L-infinity Counterexample
• Borel master codes for null and meagre sets Definition
• Conditional law given a random element Definition
• Direct integral of a measurable Hilbert field Definition
• A positive-measure compact set can miss part of every interval Example
• 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
• Empirical laws of a finite valued iid sample Example
• For every positive ε there is a dense open subset of (0,1) of Lebesgue measure below ε Example
• Likelihood ratio martingale Example
• Mollification of a complex two-step function Example
• Strong law for empirical indicator averages Example
• The Dirichlet function satisfies Lusin's conclusion without being continuous anywhere Example
• The graph of a continuous function ℝ→ℝ is Lebesgue null in ℝ² Example
• Birkhoff's theorem requires integrability False statement
• False: measure-preserving transformations are invertible False statement
• A C¹ diffeomorphism maps Lebesgue null sets to Lebesgue null sets 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
• Assuming countable choice, simple approximants to a measurable function can be made uniformly convergent on a large closed set Lemma
• Assuming countable choice, simple functions are continuous on a large closed core Lemma
• Basic identities for a probability measure Lemma
• Cauchy sequences in probability have a measurable limit Lemma
• Chacon partial maps extend to an invertible map mod null sets Lemma
• Complex Lq norm recovery from finite simple dual tests Lemma
• Conditional expectation process is a martingale Lemma
• Countable boundary null partitions of a separable metric space Lemma
• Diagonal multipliers form a von Neumann algebra Lemma
• Finite-measure growth increment lemma Lemma
• Hedberg pointwise inequality for Riesz potentials Lemma

…and 59 more results.

Dependency tree · two levels

16 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 2.58 (standard reference, not scraped)