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 [math]\displaystyle{ \mu }[/math] on measurable space [math]\displaystyle{ (X, \mathcal A) }[/math] is called a finite measure if it satisfies
- [math]\displaystyle{ \mu(X) \lt \infty. }[/math]
By the monotonicity of measures, this implies
- [math]\displaystyle{ \mu(A) \lt \infty \text{ for all } A \in \mathcal A. }[/math]
If [math]\displaystyle{ \mu }[/math] is a finite measure, the measure space [math]\displaystyle{ (X, \mathcal A, \mu) }[/math] 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 [math]\displaystyle{ X }[/math] is a Hausdorff space and [math]\displaystyle{ \mathcal A }[/math] contains the Borel [math]\displaystyle{ \sigma }[/math]-algebra then every finite measure is also a locally finite Borel measure.
Metric spaces
If [math]\displaystyle{ X }[/math] is a metric space and the [math]\displaystyle{ \mathcal A }[/math] is again the Borel [math]\displaystyle{ \sigma }[/math]-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 [math]\displaystyle{ X }[/math]. The weak topology corresponds to the weak* topology in functional analysis. If [math]\displaystyle{ X }[/math] is also separable, the weak convergence is metricized by the Lévy–Prokhorov metric.[2]
Polish spaces
If [math]\displaystyle{ X }[/math] is a Polish space and [math]\displaystyle{ \mathcal A }[/math] is the Borel [math]\displaystyle{ \sigma }[/math]-algebra, then every finite measure is a regular measure and therefore a Radon measure.[3] If [math]\displaystyle{ X }[/math] is Polish, then the set of all finite measures with the weak topology is Polish too.[4]
References
- ↑ 1.0 1.1 Hazewinkel, Michiel, ed. (2001), "Measure space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Measure_space
- ↑ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 252. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_341.
- ↑ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 248. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_646.
- ↑ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. 77. Switzerland: Springer. p. 112. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
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