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 ( X , A ) {\displaystyle (X,{\mathcal {A}})} be a measurable space and μ {\displaystyle \mu } a measure on this measurable space. The measure μ {\displaystyle \mu } is called an s-finite measure, if it can be written as a countable sum of finite measures ν n {\displaystyle \nu _{n}} ( n ∈ N {\displaystyle n\in \mathbb {N} } ),
Example The Lebesgue measure λ {\displaystyle \lambda } is an s-finite measure. For this, set
and define the measures ν n {\displaystyle \nu _{n}} by
for all measurable sets A {\displaystyle A} . These measures are finite, since ν n ( A ) ≤ ν n ( B n ) = 2 {\displaystyle \nu _{n}(A)\leq \nu _{n}(B_{n})=2} for all measurable sets A {\displaystyle 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 μ {\displaystyle \mu } be σ-finite. Then there are measurable disjoint sets B 1 , B 2 , … {\displaystyle B_{1},B_{2},\dots } with μ ( B n ) < ∞ {\displaystyle \mu (B_{n})<\infty } and
Then the measures
are finite and their sum is μ {\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 X = { a } {\displaystyle X=\{a\}} with the σ-algebra A = { { a } , ∅ } {\displaystyle {\mathcal {A}}=\{\{a\},\emptyset \}} . For all n ∈ N {\displaystyle n\in \mathbb {N} } , let ν n {\displaystyle \nu _{n}} be the counting measure on this measurable space and define
The measure μ {\displaystyle \mu } is by construction s-finite (since the counting measure is finite on a set with one element). But μ {\displaystyle \mu } is not σ-finite, since
So μ {\displaystyle \mu } cannot be σ-finite.
Equivalence to probability measures For every s-finite measure μ = ∑ n = 1 ∞ ν n {\displaystyle \mu =\sum _{n=1}^{\infty }\nu _{n}} , there exists an equivalent probability measure P {\displaystyle P} , meaning that μ ∼ P {\displaystyle \mu \sim P} . One possible equivalent probability measure is given by
References
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.
