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.[1] The theorem is due to Dorothy Maharam. It was extended to localizable measure spaces by Irving Segal.[2]

The result is important to classical Banach space theory, in that, when considering the Banach space given as an L_p 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 algebras. 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 of discrete abelian von Neumann algebras; that is, algebras of bounded functions on a discrete set.

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.

References

Maharam, Dorothy (1942). "On homogeneous measure algebras". Proceedings of the National Academy of Sciences of the United States of America. 28 (3): 108–111. doi:10.1073/pnas.28.3.108. JSTOR 87851. PMC 1078424. PMID 16578030.
Segal, Irving E. (1951). "Equivalences of measure spaces". American Journal of Mathematics. 73 (2): 275–313. doi:10.2307/2372178. JSTOR 2372178.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Maharam, Dorothy (1942). "On homogeneous measure algebras". Proceedings of the National Academy of Sciences of the United States of America. 28 (3): 108–111. doi:10.1073/pnas.28.3.108. JSTOR 87851. PMC 1078424. PMID 16578030.

[2] Segal, Irving E. (1951). "Equivalences of measure spaces". American Journal of Mathematics. 73 (2): 275–313. doi:10.2307/2372178. JSTOR 2372178.

Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics