This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(May 2022) (Learn how and when to remove this message)

Inmeasure theory, a branch of mathematics that studies generalized notions of volumes, ans-finite measure is a special type ofmeasure. 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 theσ-finite (sigma-finite) measures.

Let${\displaystyle (X,\mathcal{A})}$ be ameasurable space and${\displaystyle \mu }$ a measure on this measurable space. The measure${\displaystyle \mu }$ is called an s-finite measure, if it can be written as acountable sum offinite measures${\displaystyle {\nu }_n}$ (${\displaystyle n\in \mathbb{N}}$),^[1]

${\displaystyle \mu =\sum _{n=1}^{\mathrm{\infty }}{\nu }_n.}$

TheLebesgue measure${\displaystyle \lambda }$ is an s-finite measure. For this, set

${\displaystyle B_n=(-n,-n+1]\cup [n-1,n)}$

and define the measures${\displaystyle {\nu }_n}$ by

${\displaystyle {\nu }_n(A)=\lambda (A\cap B_n)}$

for all measurable sets${\displaystyle A}$. These measures are finite, since${\displaystyle {\nu }_n(A)\le {\nu }_n(B_n)=2}$ for all measurable sets${\displaystyle A}$, and by construction satisfy

${\displaystyle \lambda =\sum _{n=1}^{\mathrm{\infty }}{\nu }_n.}$

Therefore the Lebesgue measure is s-finite.

Everyσ-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${\displaystyle \mu }$ be σ-finite. Then there are measurable disjoint sets${\displaystyle B_1,B_2,\dots }$ with and

${\displaystyle \bigcup _{n=1}^{\mathrm{\infty }}{B}_n=X}$

Then the measures

${\displaystyle {\nu }_n(\cdot ):=\mu (\cdot \cap B_n)}$

are finite and their sum is${\displaystyle \mu }$. 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${\displaystyle X=\{a\}}$ with theσ-algebra${\displaystyle \mathcal{A}=\{\{a\},\mathrm{\varnothing }\}}$. For all${\displaystyle n\in \mathbb{N}}$, let${\displaystyle {\nu }_n}$ be thecounting measure on this measurable space and define

${\displaystyle \mu :=\sum _{n=1}^{\mathrm{\infty }}{\nu }_n.}$

The measure${\displaystyle \mu }$ is by construction s-finite (since the counting measure is finite on a set with one element). But${\displaystyle \mu }$ is not σ-finite, since

${\displaystyle \mu (\{a\})=\sum _{n=1}^{\mathrm{\infty }}{\nu }_n(\{a\})=\sum _{n=1}^{\mathrm{\infty }}1=\mathrm{\infty }.}$

So${\displaystyle \mu }$ cannot be σ-finite.

For every s-finite measure${\displaystyle \mu =\sum _{n=1}^{\mathrm{\infty }}{\nu }_n}$, there exists anequivalentprobability measure${\displaystyle P}$, meaning that${\displaystyle \mu \sim P}$.^[1] One possible equivalent probability measure is given by

${\displaystyle P=\sum _{n=1}^{\mathrm{\infty }}{2}^{-n}\frac{{\nu }_n}{{\nu }_n(X)}.}$

• 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.

• Measures (measure theory)

• Articles with short description
• Short description is different from Wikidata
• Articles lacking in-text citations from May 2022
• All articles lacking in-text citations