nLab measure theory




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.

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.

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

  • 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

Via topos theory

Discussion via topos theory

and particularly via Boolean toposes:

Last revised on May 4, 2024 at 07:28:00. See the history of this page for a list of all contributions to it.