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 ${\displaystyle \mu }$ on measurable space ${\displaystyle (X,{\mathcal {A}})}$ is called a finite measure if it satisfies

${\displaystyle \mu (X)<\infty .}$

By the monotonicity of measures, this implies

${\displaystyle \mu (A)<\infty {\text{ for all }}A\in {\mathcal {A}}.}$

If ${\displaystyle \mu }$ is a finite measure, the measure space ${\displaystyle (X,{\mathcal {A}},\mu )}$ 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 ${\displaystyle X}$ is a Hausdorff space and ${\displaystyle {\mathcal {A}}}$ contains the Borel ${\displaystyle \sigma }$-algebra then every finite measure is also a locally finite Borel measure.

Metric spaces

If ${\displaystyle X}$ is a metric space and the ${\displaystyle {\mathcal {A}}}$ is again the Borel ${\displaystyle \sigma }$-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 ${\displaystyle X}$. The weak topology corresponds to the weak* topology in functional analysis. If ${\displaystyle X}$ is also separable, the weak convergence is metricized by the Lévy–Prokhorov metric.^[2]

Polish spaces

If ${\displaystyle X}$ is a Polish space and ${\displaystyle {\mathcal {A}}}$ is the Borel ${\displaystyle \sigma }$-algebra, then every finite measure is a regular measure and therefore a Radon measure.^[3] If ${\displaystyle X}$ is Polish, then the set of all finite measures with the weak topology is Polish too.^[4]

References

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