Measure theory studies measurable spaces and measure spaces.
Measure theory is the field of mathematics that grew out of the Lebesgue integral and Kolmogorov's axioms for probability.
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.
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).
A comprehensive five-volume treatise (with a sixth volume forthcoming) is
A more concise two-volume treatise is
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.
Discussion via topos theory
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)
There is an approach to measure theory using $\sigma$-locales:
which is part of Simpson’s work to develop synthetic probability theory.
Last revised on August 29, 2024 at 21:34:38. See the history of this page for a list of all contributions to it.