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. Let be a measurable space and a measure on this measurable space. The measure is called an s-finite measure, if it can be written as a countable sum of finite measures (), 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 be σ-finite. Then there are measurable disjoint sets with and Then the measures are finite and their sum is . 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 with the σ-algebra . For all , let be the counting measure on this measurable space and define The measure is by construction s-finite (since the counting measure is finite on a set with one element). But is not σ-finite, since So cannot be σ-finite. For every s-finite measure , there exists an equivalent probability measure , meaning that .