Measure algebra

Definition

A measure algebra is a pair $(\mathfrak{A}, \mu)$ such that:

$\mathfrak{A}$ is a Boolean algebra in which every countable subset has a supremum.
$\mu \colon \mathfrak{A} \to [0, \infty]$ satisfies $\mu(0) = 0$ .
$\mu\left(\bigvee_{k} a_k\right) = \sum_k \mu(a_k)$ for every sequence $(a_k)_{k \in \N}$ of pairwise disjoint elements, that is, $a_j \wedge a_k = 0$ for $j \ne k$ .
$\mu(a) > 0$ for every $a \ne 0$ .

Remarks

Example: let $(X, \Sigma, \nu)$ be a measure space and $\mathcal{N} = \set{E \in \Sigma : \nu(E) = 0}$ . Call $E, F \in \Sigma$ equivalent when $\nu(E \mathbin{\triangle} F) = 0$ . The classes form a Boolean algebra $\Sigma / \mathcal{N}$ with countable suprema given by countable unions, and $\mu([E]) = \nu(E)$ is well defined. Clauses 2 and 3 come from the measure $\nu$ , and clause 4 holds since $[E] \ne 0$ means $\nu(E) > 0$ .

Non-example: the sigma-algebra of Borel subsets of $\R$ with Lebesgue measure satisfies clauses 1 to 3 but not 4. The set $\set{0}$ is a nonzero element of measure $0$ .

The quotient in the example is called the measure algebra of $(X, \Sigma, \nu)$ .

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Boolean algebra concept
Measure concept
Measure algebra definition

Tag	`KST0YNAP55`
URI	`m2://KST0YNAP55`
Kind	definition
Writing	W0 stub
MSC	28A60, 06E10
Defines	Measure algebra
Uses	Boolean algebra, Measure
Changed	2 November 2026

Measure algebra

Definition

Remarks

Reading path

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