Measure space

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 ) , {\displaystyle (X,{\mathcal {A}},\mu ),} where[1][2]

  • X {\displaystyle X} is a set
  • A {\displaystyle {\mathcal {A}}} is a σ-algebra on the set X {\displaystyle X}
  • μ {\displaystyle \mu } is a measure on ( X , A ) {\displaystyle (X,{\mathcal {A}})}

In other words, a measure space consists of a measurable space ( X , A ) {\displaystyle (X,{\mathcal {A}})} together with a measure on it.

Example

Set X = { 0 , 1 } {\displaystyle X=\{0,1\}} . The σ {\textstyle \sigma } -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 ℘ ( ⋅ ) . {\textstyle \wp (\cdot ).} Sticking with this convention, we set

A = ℘ ( X ) {\displaystyle {\mathcal {A}}=\wp (X)}

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

℘ ( X ) = { ∅ , { 0 } , { 1 } , { 0 , 1 } } . {\displaystyle \wp (X)=\{\varnothing ,\{0\},\{1\},\{0,1\}\}.}

As the measure, define μ {\textstyle \mu } by

μ ( { 0 } ) = μ ( { 1 } ) = 1 2 , {\displaystyle \mu (\{0\})=\mu (\{1\})={\frac {1}{2}},}
so μ ( X ) = 1 {\textstyle \mu (X)=1} (by additivity of measures) and μ ( ∅ ) = 0 {\textstyle \mu (\varnothing )=0} (by definition of measures).

This leads to the measure space ( X , ℘ ( X ) , μ ) . {\textstyle (X,\wp (X),\mu ).} It is a probability space, since μ ( X ) = 1. {\textstyle \mu (X)=1.} The measure μ {\textstyle \mu } corresponds to the Bernoulli distribution with p = 1 2 , {\textstyle p={\frac {1}{2}},} 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:

  • Probability spaces, a measure space where the measure is a probability measure[1]
  • Finite measure spaces, where the measure is a finite measure[3]
  • σ {\displaystyle \sigma } -finite measure spaces, where the measure is a σ {\displaystyle \sigma } -finite measure[3]

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

References

  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.
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