Measures

Last modified: 2026-11-03 15:17:03 (PST)

Remark. A measure assigns a size to each set in a collection of sets: how many elements it has, how long it is, or how likely it is. These notes build measures from \(\sigma\)-algebras and additivity, using the definitions on the Sets and Functions pages. One kind of measure, the probability measure (Definition 12 in Measures and Probability Measures), is the starting point of the Morrison Lab’s probability notes (probability measure).

The measure notes are split across these pages:

  • Sigma-Algebras: \(\sigma\)-algebras, generated \(\sigma\)-algebras, and the Borel \(\sigma\)-algebra.
  • Additivity: finite and countable additivity.
  • Measures and Probability Measures: measures, the counting and Lebesgue measures, measure spaces, null sets, \(\sigma\)-finite measures, probability measures, and probability spaces.
  • Measurable Functions and Integrals: measurable functions, the integral, and absolutely integrable functions.
  • \(L^p\) Spaces: \(L^p\) norms and spaces, the essential supremum, and the Hölder and Minkowski inequalities.
  • Product Measures: product \(\sigma\)-algebras and product measures.

A link to a section, theorem or definition on this page from before the split goes on to the page that now holds it.

1 Further reading

  • Billingsley [1] develops measure theory and builds probability on it. Its early chapters define \(\sigma\)-algebras, measures, and their basic properties.
  • Folland [2] is a graduate text on measure and integration; its first chapters treat \(\sigma\)-algebras and measures.
  • Gut [3] is a graduate course in probability that opens with a chapter on introductory measure theory.

References

[1]
P. Billingsley, Probability and measure, 3rd ed. in Wiley series in probability and mathematical statistics. New York: Wiley, 1995.
[2]
G. B. Folland, Real analysis: Modern techniques and their applications, 2nd ed. Wiley, 1999. Available: https://www.wiley.com/en-us/Real+Analysis%3A+Modern+Techniques+and+Their+Applications%2C+2nd+Edition-p-9780471317166
[3]
A. Gut, Probability: A graduate course, 2nd ed. in Springer texts in statistics. New York: Springer, 2013. doi: 10.1007/978-1-4614-4708-5.