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.
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.
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.
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.
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.