Finite measure

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

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]

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]