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Concept
Finite measure
Summary
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.
A measure on measurable space is called a finite measure if it satisfies
By the monotonicity of measures, this implies
If is a finite measure, the measure space is called a finite measure space or a totally finite measure space.
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.
If is a Hausdorff space and contains the Borel -algebra then every finite measure is also a locally finite Borel measure.
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.
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