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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.
By the monotonicity of measures, this implies
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.
If is a metric space and the is again the Borel -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 . The weak topology corresponds to the weak* topology in functional analysis. If is also separable, the weak convergence is metricized by the Lévy–Prokhorov metric.
If is a Polish space and is the Borel -algebra, then every finite measure is a regular measure and therefore a Radon measure. If is Polish, then the set of all finite measures with the weak topology is Polish too.
- Anosov, D.V. (2001) , "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. Switzerland: Springer. p. 112. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
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