s-finite measure

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.

be a measurable space and

is called an s-finite measure, if it can be written as a countable sum of finite measures

),[1] The Lebesgue measure

is an s-finite measure.

For this, set and define the measures

by for all measurable sets

These measures are finite, since

for all measurable sets

, and by construction satisfy Therefore the Lebesgue measure is s-finite.

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

Then there are measurable disjoint sets

and Then the measures are finite and their sum is

An example for an s-finite measure that is not σ-finite can be constructed on the set

is by construction s-finite (since the counting measure is finite on a set with one element).

For every s-finite measure

, there exists an equivalent probability measure

[1] One possible equivalent probability measure is given by