Finite measure

Inmeasure theory,a branch ofmathematics,afinite measureortotally finite measure^[1]is a specialmeasurethat always takes on finite values. Among finite measures areprobability measures.The finite measures are often easier to handle than more general measures and show a variety of different properties depending on thesetsthey are defined on.

Definition

Ameasure $\mu$ onmeasurable space $(X,{\mathcal {A}})$ is called a finite measure if it satisfies

\mu (X)<\infty.

By the monotonicity of measures, this implies

\mu (A)<\infty {\text{ for all }}A\in {\mathcal {A}}.

If $\mu$ is a finite measure, themeasure space $(X,{\mathcal {A}},\mu )$ is called afinite measure spaceor atotally finite measure space.^[1]

Properties

General case

For any measurable space, the finite measures form aconvex conein theBanach spaceofsigned measureswith thetotal variationnorm. Important subsets of the finite measures are the sub-probability measures, which form aconvex subset,and the probability measures, which are the intersection of theunit spherein the normed space of signed measures and the finite measures.

Topological spaces

If $X$ is aHausdorff spaceand ${\mathcal {A}}$ contains theBorel $\sigma$ -algebrathen every finite measure is also alocally finite Borel measure.

Metric spaces

If $X$ is ametric spaceand the ${\mathcal {A}}$ is again the Borel $\sigma$ -algebra, theweak convergence of measurescan be defined. The corresponding topology is called weak topology and is theinitial topologyof all bounded continuous functions on $X$ .The weak topology corresponds to theweak* topologyin functional analysis. If $X$ is alsoseparable,the weak convergence is metricized by theLévy–Prokhorov metric.^[2]

Polish spaces

If $X$ is aPolish spaceand ${\mathcal {A}}$ is the Borel $\sigma$ -algebra, then every finite measure is aregular measureand therefore aRadon measure.^[3] If $X$ is Polish, then the set of all finite measures with the weak topology is Polish too.^[4]

References

^^a ^bAnosov, D.V. (2001) [1994],"Measure space",Encyclopedia of Mathematics,EMS Press
^Klenke, Achim (2008).Probability Theory.Berlin: Springer. p.252.doi:10.1007/978-1-84800-048-3.ISBN 978-1-84800-047-6.
^Klenke, Achim (2008).Probability Theory.Berlin: Springer. p.248.doi:10.1007/978-1-84800-048-3.ISBN 978-1-84800-047-6.
^Kallenberg, Olav(2017).Random Measures, Theory and Applications.Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 112.doi:10.1007/978-3-319-41598-7.ISBN 978-3-319-41596-3.

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[eommeasurespace-1] Anosov, D.V. (2001) [1994],"Measure space",Encyclopedia of Mathematics,EMS Press

[Klenke252-2] Klenke, Achim (2008).Probability Theory.Berlin: Springer. p.252.doi:10.1007/978-1-84800-048-3.ISBN 978-1-84800-047-6.

[Klenke248-3] Klenke, Achim (2008).Probability Theory.Berlin: Springer. p.248.doi:10.1007/978-1-84800-048-3.ISBN 978-1-84800-047-6.

[Kallenberg112-4] Kallenberg, Olav(2017).Random Measures, Theory and Applications.Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 112.doi:10.1007/978-3-319-41598-7.ISBN 978-3-319-41596-3.

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