Measure space

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Short description: Set on which a generalization of volumes and integrals is defined

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.

Definition

A measure space is a triple (X,𝒜,μ), where[1][2]

In other words, a measure space consists of a measurable space (X,𝒜) together with a measure on it.

Example

Set X={0,1}. The σ-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 𝒜=℘(X)

In this simple case, the power set can be written down explicitly: ℘(X)={∅,{0},{1},{0,1}}.

As the measure, define μ by μ({0})=μ({1})=12, so μ(X)=1 (by additivity of measures) and μ(∅)=0 (by definition of measures).

This leads to the measure space (X,℘(X),μ). It is a probability space, since μ(X)=1. The measure μ corresponds to the Bernoulli distribution with p=12, 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]

References

  1. ↑ 1.0 1.1 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. ↑ 3.0 3.1 Hazewinkel, Michiel, ed. (2001), "Measure space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Measure_space 
  4. ↑ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 33. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. 



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