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


  1. 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

[1]

[2]

[3]

[4]

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