Maharam's theorem

In mathematics, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach spaces.

In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of products of the unit interval [0,1] on the reals), and "purely atomic parts," using the counting measure on some discrete space.

It was extended to localizable measure spaces by Irving Segal.

Maharam's theorem can also be translated into the language of abelian von Neumann algebras.

A similar theorem was given by Kazimierz Kuratowski for Polish spaces, stating that they are isomorphic, as Borel spaces, to either the reals, the integers, or a finite set.