Measure space


A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ-algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space.

A measurable space consists of the first two components without a specific measure.



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

  •   is a set
  •   is a σ-algebra on the set  
  •   is a measure on  

In other words, a measure space consists of a measurable space   together with a measure on it.



Set  . 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 the 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


Most important classes of measure spaces are defined by the properties of their associated measures. This includes, in order of increasing generality:

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", Encyclopedia of Mathematics, EMS Press
  4. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 33. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.