Finite measure
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.

Definition 
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.

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  is a Hausdorff space and  contains the Borel -algebra then every finite measure is also a locally finite Borel measure.

Metric spaces 
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.

Polish spaces 
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.

References 

Measures (measure theory)