Finite measure explained

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

\mu on measurable space (X, l A) is called a finite measure if it satisfies\mu(X) < inf ty.

By the monotonicity of measures, this implies

\mu(A) < inf ty forall A \in l A.

If

\mu is a finite measure, the measure space (X, l 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 l 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 l 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 l 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

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Notes and References

1. Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling . 77 . Switzerland . Springer . 112. 10.1007/978-3-319-41598-7. 978-3-319-41596-3.
2. Book: Klenke . Achim . 2008 . Probability Theory . registration . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6 . 248.
3. Book: Klenke . Achim . 2008 . Probability Theory . registration . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6 . 252.