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.
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.
is called a finite measure if it satisfies By the monotonicity of measures, this implies If
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.
-algebra then every finite measure is also a locally finite Borel measure.
-algebra, the weak convergence of measures can be defined.
The weak topology corresponds to the weak* topology in functional analysis.
is Polish, then the set of all finite measures with the weak topology is Polish too.
[4] This mathematical analysis–related article is a stub.