Maharam's theorem
Encyclopedia
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 space
s. In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of the unit interval
[0,1] on the reals), and "purely atomic parts", using the counting measure
on some discrete space. The theorem is due to Dorothy Maharam
.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s. In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of the unit interval
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...
[0,1] on the reals), and "purely atomic parts", using the counting measure
Counting measure
In mathematics, 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 is finite, and ∞ if the subset is infinite....
on some discrete space. The theorem is due to Dorothy Maharam
Dorothy Maharam
Dorothy Maharam Stone is an American mathematician who made important contributions to measure theory. Her husband was British mathematician Arthur Harold Stone....
.
The result is important to classical Banach space theory, in that, when considering the Banach space given as an Lp space
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
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.