Measure space

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A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. Measure spaces contain information about the underlying set, the subsets of said set that are feasible for measuring (the -algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space.

Measure space should not be confused with the related measurable spaces.


A measure space is a triple , where[1][2]

  • is a (nonempty) set
  • is a -algebra on the set
  • is a measure on



The -algebra on finite sets such as the one above is usually the power set, which is the set of all subsets (of a given set) and is denoted by . Sticking with this convention, we set

In this simple case, the power set can be written down explicitly:

As measure, define by

so (by additivity of measures) and (by definition of measures).

This leads to the measure space . It is a probability space, since . The measure corresponds to the Bernoulli distribution with , which is for example used to model a fair coin flip.

Important classes of measure spaces[edit]

Most important classes of measure spaces are defined by the properties of their associated measures. This includes

Another class of measure spaces are the complete measure spaces.[4]


  1. ^ a b Kosorok, Michael R. (2008). Introduction to Empirical Processes and Semiparametric Inference. New York: Springer. p. 83. ISBN 978-0-387-74977-8.
  2. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 18. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  3. ^ a b Anosov, D.V. (2001) [1994], "Measure space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  4. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 33. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.