# 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.

## Definition

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 (),[1]

## Example

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.

## 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 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.

### Equivalence to probability measures

For every s-finite measure , there exists an equivalent probability measure , meaning that .[1] One possible equivalent probability measure is given by

Here, the are finite measures that sum up to like in the definition.

## References

- Kallenberg, Olav (2017).
*Random Measures, Theory and Applications*. Switzerland: Springer. p. 21. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.

Sources for s-finite measures

- Falkner, Neil (2009). "Reviews".
*American Mathematical Monthly*.**116**(7): 657–664. doi:10.4169/193009709X458654. ISSN 0002-9890. - Olav Kallenberg (12 April 2017).
*Random Measures, Theory and Applications*. Springer. ISBN 978-3-319-41598-7. - Günter Last; Mathew Penrose (26 October 2017).
*Lectures on the Poisson Process*. Cambridge University Press. ISBN 978-1-107-08801-6. - R.K. Getoor (6 December 2012).
*Excessive Measures*. Springer Science & Business Media. ISBN 978-1-4612-3470-8.