mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Maharam's theorem is a deep result about the decomposability of

measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...

s, which plays an important role in the theory of

Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...

s. In brief, it states that every

complete measure space In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (''X'', Σ, ''μ'') is compl ...

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 analysi ...

,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 infinit ...

on some discrete space. The theorem is due to Dorothy Maharam. It was extended to localizable measure spaces by

Irving Segal Irving Ezra Segal (1918–1998) was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is often referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for h ...

. The result is important to classical Banach space theory, in that, when considering the Banach space given as an L_p space of

measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...

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 In functional analysis, a branch of mathematics, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commutative, commute. The prototypical example of an abelian von Neumann algebra is th ...

s. Every abelian von Neumann algebra is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

to a product of abelian von Neumann algebras, and every abelian von Neumann algebra is isomorphic to a spatial

tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...

of discrete abelian von Neumann algebras; that is, algebras of

bounded function In mathematics, a function f defined on some set X with real or complex values is called bounded if the set of its values (its image) is bounded. In other words, there exists a real number M such that :, f(x), \le M for all x in X. A functi ...

s on a

discrete set In mathematics, a point (topology), point is called an isolated point of a subset (in a topological space ) if is an element of and there exists a Neighborhood (mathematics), neighborhood of that does not contain any other points of . This i ...

. 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. He worked as a professor at the University of Warsaw and at the Ma ...

for

Polish space In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...

s, stating that they are isomorphic, as Borel spaces, to either the reals, the

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s, or a finite set.

References

Banach spaces Theorems in measure theory {{mathanalysis-stub