Measure and integration Measure-preserving maps and further topics Theorem

Isomorphism theorem for measure algebras

Builds on Boolean algebra Lebesgue measure Null set

Statement

Let $\mathfrak{A}$ be a Boolean algebra in which every countable subset has a supremum, and let $m \colon \mathfrak{A} \to [0, 1]$ . The pair $(\mathfrak{A}, m)$ is a probability measure algebra if three conditions hold.

$m(1) = 1$ , and $m(a) = 0$ only for $a = 0$ .
$m\left(\bigvee_k a_k\right) = \sum_k m(a_k)$ for every sequence of pairwise disjoint elements $a_k$ .
The set $\mathfrak{A}$ is separable for the metric $d(a, b) = m(a \mathbin{\triangle} b)$ .

Assume also that $(\mathfrak{A}, m)$ is atomless: every $a \ne 0$ has some $b \le a$ with $0 < m(b) < m(a)$ . Let $\mathfrak{L}$ be the algebra of Lebesgue measurable subsets of $[0,1]$ modulo null sets, with $\lambda$ induced by Lebesgue measure. Then there is a Boolean algebra isomorphism $\Phi \colon \mathfrak{A} \to \mathfrak{L}$ with $\lambda(\Phi(a)) = m(a)$ for all $a \in \mathfrak{A}$ .

Remarks

Idea of proof: take a dense sequence in $\mathfrak{A}$ and the finite partitions of $1$ that it generates. Since the algebra is atomless, $m$ takes all values between $0$ and $m(a)$ on elements below $a$ . This lets one send each partition to a partition of $[0,1]$ into finite unions of intervals with the same measures, compatibly with refinement. The map then extends by continuity in the metric $d$ . Without atomlessness the result fails: the algebra with two atoms of measure $1/2$ has four elements, while $\mathfrak{L}$ is infinite. Without separability it fails as well: the measure algebra of the product measure on $\set{0,1}^I$ with $I$ uncountable is atomless but not separable. As a consequence, the measure algebra of every atomless standard probability space is isomorphic to that of $[0,1]$ .

Going deeper

For a student

The theorem lets ergodic theory move between point maps and automorphisms of measure algebras. Together with the theory of standard probability spaces, it shows that many probability spaces of analysis are isomorphic to $[0,1]$ up to null sets. An example is $[0,1]^\N$ with the product of Lebesgue measures. Maharam's theorem classifies the non-separable case.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Boolean algebra concept
Borel sigma-algebra concept
Euclidean space object
Null set concept
Outer measure concept
Lebesgue measure object
Isomorphism theorem for measure algebras statement

Tag	`YFAQGE2ATQ`
URI	`m2://YFAQGE2ATQ`
Kind	statement, theorem
Writing	W0 stub
MSC	28A60, 28D05
Status	This copy of the site has no build data. Run `m2 status YFAQGE2ATQ`.
About	Lebesgue measure
Uses	Boolean algebra, Lebesgue measure, Null set
Changed	2 November 2026