Measures on sigma-algebras

Depends on

• Sigma-algebras
• Series in the nonnegative extended real line
• The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined

Used by

• Every subset of ℝ of positive Lebesgue outer measure contains a nonmeasurable subset Corollary
• Polar integration may discard the cut locus Corollary
• The Solovay model has no Vitali or Bernstein set Corollary
• A non-sigma-finite premeasure has distinct Borel extensions Counterexample
• No normalized translation-invariant Haar probability on the real line Counterexample
• The zero-countable / infinity-cocountable measure space breaks the p=1 endpoint of duality Counterexample
• A Borel measure on ℝ that is finite on compact sets Definition
• A positive, signed, or complex measure concentrated on a measurable set Definition
• Absolute continuity of a signed or complex measure with respect to a positive measure Definition
• An atom of a measure on ℝ Definition
• Banach-valued vector measure and variation Definition
• Borel master codes for null and meagre sets Definition
• Finite, sigma-finite, and semifinite measures Definition
• Measure kernel and probability kernel Definition
• Measure spaces Definition
• Nonnegative scalar multiples and countable weighted sums of measures Definition
• PMEA and PMEA-sigma Definition
• Probability measures and probability spaces Definition
• Radon measure on an LCH space Definition
• Restriction of a measure to a measurable set Definition
• Support of a finite Borel measure on the plane Definition
• The distribution function of absolute value Definition
• The integral of a nonnegative simple function Definition
• The one-dimensional torus and its normalized Haar integral Definition
• A boundary atom gives an h1 function without an L1 density Example
• A square-integrable kernel without a continuous representative Example
• Arcsine equilibrium measure and capacity of a segment Example
• Assuming countable choice, zero on countable sets and infinity on cocountable sets is a non-semifinite measure Example
• Green kernel of a biased integer walk Example
• Isotypic Haar projections specialize to finite character sums Example
• Negative drift gives a finite mean small-set hit Example
• Weighted counting in dimension zero Example
• FALSE: a measure on an infinite set that vanishes on every singleton is the zero measure False statement
• FALSE: every finitely additive nonnegative function on an algebra extends to a measure False statement
• FALSE: every finitely additive nonnegative set function on a sigma-algebra is a measure False statement
• FALSE: every subset of a measure-null set is measurable False statement
• FALSE: measures are additive on arbitrary countable unions False statement
• FALSE: the extension of a premeasure is always unique False statement
• A measurable set of positive finite measure occupies more than any prescribed proportion of some dyadic cube Lemma
• A shear sends the unit cube to a set of Lebesgue measure one Lemma

…and 51 more results.

Dependency tree · two levels

12 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, §2C (standard reference, not scraped)
• G. Folland, Real Analysis, 2nd ed., §1.3 (standard reference, not scraped)