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 ${\displaystyle \mu }$ on measurable space ${\displaystyle (X,{\mathcal {A}})}$ is called a finite measure iff it satisfies

${\displaystyle \mu (X)<\infty .}$

By the monotonicity of measures, this implies

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

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

Metric spaces

If ${\displaystyle X}$ is a metric space and the ${\displaystyle {\mathcal {A}}}$ is again the Borel ${\displaystyle \sigma }$-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 ${\displaystyle X}$. The weak topology corresponds to the weak* topology in functional analysis. If ${\displaystyle X}$ is also separable, the weak convergence is metricized by the Lévy–Prokhorov metric.[2]

Polish spaces

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

References

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