View source for Maharam's theorem

{{Short description|Mathematical theorem regarding decomposability of measure spaces}}
In [[mathematics]], '''Maharam's theorem''' is a deep result about the decomposability of [[measure space]]s, which plays an important role in the theory of [[Banach space]]s.  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 [[real number|reals]]), and "purely atomic parts," using the [[counting measure]] on some discrete space.<ref>{{cite journal|first=Dorothy|last= Maharam|authorlink=Dorothy Maharam|title= On homogeneous measure algebras|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|volume=28|year= 1942|issue= 3|pages=108–111|doi= 10.1073/pnas.28.3.108|jstor=87851|pmid= 16578030|pmc= 1078424|doi-access= free|bibcode= 1942PNAS...28..108M}}</ref> The theorem is due to [[Dorothy Maharam]].
It was extended to localizable measure spaces by [[Irving Segal]].<ref>{{cite journal|first=Irving E.|last= Segal|authorlink=Irving Segal |title=Equivalences of measure spaces|journal=[[American Journal of Mathematics]]|volume=73|year=1951|issue= 2|pages= 275–313|doi=10.2307/2372178|jstor=2372178}}</ref>

The result is important to classical Banach space theory, in that, when considering the Banach space given as an [[Lp space|L<sub>p</sub> space]] of [[measurable function]]s over a general measurable space, it is sufficient to understand it in terms of its decomposition into non-atomic and atomic parts.

Maharam's theorem can also be translated into the language of [[abelian von Neumann algebra]]s.  Every abelian von Neumann algebra is [[isomorphic]] to a product of {{nowrap|σ-finite}} abelian von Neumann algebras, and every {{nowrap|σ-finite}} abelian von Neumann algebra is isomorphic to a spatial [[tensor product]] of discrete abelian von Neumann algebras; that is, algebras of [[bounded function]]s on a [[discrete set]].

A similar theorem was given by [[Kazimierz Kuratowski]] for [[Polish space]]s, stating that they are isomorphic, as [[measurable space|Borel space]]s, to either the reals, the [[integer]]s, or a finite set.

==References==

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{{Banach spaces}}
{{Functional analysis}}
{{Measure theory}}

[[Category:Banach spaces]]
[[Category:Theorems in measure theory]]

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