Finite measure

In measure theory, a branch of mathematics, a finite measure or totally finite measure^[1] is a special measure that always takes on finite values. Among finite measures are probability measures. The finite measures are often easier to handle than more general measures and show a variety of different properties depending on the sets they are defined on.

Definition

A measure $\mu$ on measurable 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, the measure space $(X,{\mathcal {A}},\mu )$ is called a finite measure space or a totally finite measure space.^[1]

Properties

General case

For any measurable space, the finite measures form a convex cone in the Banach space of signed measures with the total variation norm. Important subsets of the finite measures are the sub-probability measures, which form a convex subset, and the probability measures, which are the intersection of the unit sphere in the normed space of signed measures and the finite measures.

Topological spaces

If $X$ is a Hausdorff space and ${\mathcal {A}}$ contains the Borel $\sigma$ -algebra then every finite measure is also a locally finite Borel measure.

Metric spaces

If $X$ is a metric space and the ${\mathcal {A}}$ is again the Borel $\sigma$ -algebra, the weak convergence of measures can be defined. The corresponding topology is called weak topology and is the initial topology of all bounded continuous functions on $X$ . The weak topology corresponds to the weak* topology in functional analysis. If $X$ is also separable, the weak convergence is metricized by the Lévy–Prokhorov metric.^[2]

Polish spaces

If $X$ is a Polish space and ${\mathcal {A}}$ is the Borel $\sigma$ -algebra, then every finite measure is a regular measure and therefore a Radon measure.^[3] If $X$ is Polish, then the set of all finite measures with the weak topology is Polish too.^[4]

References

^ ^a ^b Anosov, 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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