Measure space

Summary

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.

DefinitionEdit

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.

ExampleEdit

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 spacesEdit

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]

ReferencesEdit

  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.