DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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Measures on sigma-algebras

Definition

Let $X$ be a set and let $A$ be a sigma-algebra on $X$ (Sigma-algebras). A measure on $(X, A)$ is a function

$μ : A ⟶ [0, + \infty] \subseteq \overline{R}$

such that:

$μ (\emptyset) = 0$ ;
for every pairwise disjoint sequence $(E_{k})_{k \in N}$ in $A$ , $μ (⋃_{k \in N} E_{k}) = \sum_{k = 0}^{\infty} μ (E_{k}),$ where the right side is the nonnegative extended sum of Series in the nonnegative extended real line.

The second condition is countable additivity. It begins at index $0$ and includes the case in which some term or the total sum is $+ \infty$ .

Depends on

Used by

Finite, sigma-finite, and semifinite measures Definition
Measure spaces Definition
Nonnegative scalar multiples and countable weighted sums of measures Definition
Probability measures and probability spaces Definition
Restriction of a measure to a measurable set Definition
Assuming countable choice, zero on countable sets and infinity on cocountable sets is a non-semifinite measure Example
FALSE: a measure on an infinite set that vanishes on every singleton is the zero 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
Finite measures agreeing on a generating pi-system and on the whole space are equal Lemma
Counting measure is a measure Proposition
Measure of a set difference when the smaller set has finite measure Proposition
Measures are monotone Proposition
The restriction of a measure to a measurable set is a measure Proposition
The two-set measure identity μ(A∪ B)+μ(A∩ B)=μ(A)+μ(B) Proposition
A measure on a finite sigma-algebra is a finite weighted sum over its atoms Theorem
Assuming countable choice, every measure space has a unique complete extension to its completion Theorem
Assuming countable choice, the semifinite part is a semifinite measure and equals the original measure exactly when it is semifinite Theorem
Continuity from below for measures Theorem
Countable additivity and continuity of finitely additive set functions Theorem
Every measure on a countable discrete space is its weighted sum of Dirac measures Theorem
Finite and countable subadditivity of measures Theorem
Nonnegative scalar multiples and countable weighted sums of measures are measures Theorem
The first Borel-Cantelli lemma for measures Theorem

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)