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
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
,1on the reals), and "purely atomic parts", using the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ...
on some discrete space. The theorem is due to
Dorothy Maharam.
It was extended to localizable
measure spaces by
Irving Segal.
The result is important to classical Banach space theory, in that, when considering the Banach space given as an
Lp space of
measurable functions 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 In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.
The prototypical example of an abelian von Neumann algebra is the algebra ''L''∞(''X'', μ) for μ a ...
s. Every abelian von Neumann algebra is isomorphic to a product of σ-finite abelian von Neumann algebras, and every σ-finite abelian von Neumann algebra is isomorphic to a spatial
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of discrete abelian von Neumann algebras; that is, algebras of
bounded functions on a
discrete set.
A similar theorem was given by
Kazimierz Kuratowski
Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.
Biography and studies
Kazimierz Kuratowski was born in Warsaw, (t ...
for
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named be ...
s, stating that they are isomorphic, as
Borel spaces, to either the reals, the integers, or a finite set.
References
Banach spaces
Theorems in measure theory
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