nLab measure theory

Idea

Measure theory studies measurable spaces and measure spaces.

History

Measure theory is the field of mathematics that grew out of the Lebesgue integral and Kolmogorov's axioms for probability.

Categories of measure theory

From the nPOV, it is desirable to have a good category for measure theory.

The article categories of measure theory provides evidence that the category of compact strictly localizable enhanced measurable spaces captures the desired features of measure theory as presented in common textbooks on real analysis.

Some subfields, related fields, and applications

The general measure theory studies general notions and constructions in measure theory, like the connection to integration, the measure spaces, derivation by measure, Caratheodory construction? and so on.

Probability theory studies special class of measures, so called probability measures which are normalized to unity.

Measure theory is very much having a central role in studying so called ergodic theory of dynamical systems.

Geometric measure theory is the geometric study of measures of subsets of Euclidean space and the measure theoretic aspects of various geometric objects, like the integration of classes of currents and their extremization properties.

There is a generalization, the noncommutative measure theory, which is more or less the study of von Neumann algebras, see Connes (1995).

Related concepts

• expectation value

• valuation (measure theory)

• noncommutative measure theory

References

General

A comprehensive five-volume treatise (with a sixth volume forthcoming) is

• David H. Fremlin, Measure Theory, Volumes 1–5, Torres Fremlin, Colchester. Volume 1, 2000; Volume 2, 2001; Volume 3, 2002; Volume 4, 2003; Volume 5, 2008. Website.

A more concise two-volume treatise is

• Vladimir Bogachev, Measure theory?, Volumes I, II. Springer, 2007. ISBN: 978-3-540-34513-8, 3-540-34513-2.

A classical (slightly dated) concise treatise is

• Paul Halmos, Measure Theory, D. Van Nostrand Company, 1950.

• Donald L. Cohn, Measure Theory, Birkhäuser, 1980. ISBN: 3-7643-3003-1

Other texts include

• Ernst-Erich Doberkat, Measures and all that — A Tutorial (arXiv:1409.2662)

• Alain Connes (1995); Noncommutative Geometry.

Via topos theory

Discussion via topos theory

• Matthew Jackson, A sheaf-theoretic approach to measure theory, 2006. (pdf, d-scholarship:7348)

and particularly via Boolean toposes:

• Simon Henry, Measure theory over boolean toposes, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 163 Issue 1, 2016 (arXiv:1411.1605, doi:10.1017/S0305004116000700)

• Asgar Jamneshan, Terence Tao, Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration (arXiv:2010.00681)