In measure theory, a branch of mathematics that studies generalized notions of volumes, an s-finite measure is a special type of measure. An s-finite measure is more general than a finite measure, but allows one to generalize certain proofs for finite measures. The s-finite measures should not be confused with the σ-finite (sigma-finite) measures.

Definition

Let be a measurable space and \mu a measure on this measurable space. The measure \mu is called an s-finite measure, if it can be written as a countable sum of finite measures \nu_n (n \in \N),

Example

The Lebesgue measure \lambda is an s-finite measure. For this, set and define the measures \nu_n by for all measurable sets A. These measures are finite, since for all measurable sets A, and by construction satisfy Therefore the Lebesgue measure is s-finite.

Properties

Relation to σ-finite measures

Every σ-finite measure is s-finite, but not every s-finite measure is also σ-finite. To show that every σ-finite measure is s-finite, let \mu be σ-finite. Then there are measurable disjoint sets with and Then the measures are finite and their sum is \mu. This approach is just like in the example above. An example for an s-finite measure that is not σ-finite can be constructed on the set X={a} with the σ-algebra. For all n \in \N, let \nu_n be the counting measure on this measurable space and define The measure \mu is by construction s-finite (since the counting measure is finite on a set with one element). But \mu is not σ-finite, since So \mu cannot be σ-finite.

Equivalence to probability measures

For every s-finite measure, there exists an equivalent probability measure P, meaning that \mu \sim P. One possible equivalent probability measure is given by