finite measure

In measure theory, a branch of mathematics, a finite measure or totally finite measure 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.

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.

= 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.

If $X$ is Polish, then the set of all finite measures with the weak topology is Polish too.

References

{{cite book |last1=Kallenberg |first1=Olav |author-link1=Olav Kallenberg |year=2017 |title=Random Measures, Theory and Applications|series=Probability Theory and Stochastic Modelling |volume=77 |location= Switzerland |publisher=Springer |page=112|doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3}}

{{cite book |last1=Klenke |first1=Achim |year=2008 |title=Probability Theory |url=https://archive.org/details/probabilitytheor00klen_646 |url-access=registration |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6 |page=[https://archive.org/details/probabilitytheor00klen_646/page/n249 248]}}

{{cite book |last1=Klenke |first1=Achim |year=2008 |title=Probability Theory |url=https://archive.org/details/probabilitytheor00klen_341 |url-access=registration |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6 |page=[https://archive.org/details/probabilitytheor00klen_341/page/n253 252]}}

{{SpringerEOM |title=Measure space |id=Measure_space |author-last1=Anosov |author-first1=D.V.}}

Category:Measures (measure theory)