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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.

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**.^{[1]}

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.^{[2]}

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