S-finite measure explained

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, l A ) be a measurable space and \mu a measure on this measurable space. The measure \mu is called an s-finite measure, if it can be written as a countable sum of finite measures \nu _{n} (n \in \N ),\mu = \sum ^{inf ty} _{n =1} \nu _{n.}

Example

λ is an s-finite measure. For this, setB _{n=} ( - n, - n +1 ] \cup [n -1 ,n) and define the measures \nu _{n} by\nu _{n(A)=} λ (A \cap B _{n)} for all measurable sets A . These measures are finite, since \nu _{n(A)} \leq \nu _{n(B} _{n)=2} for all measurable sets A , and by construction satisfyλ = \sum ^{inf ty} _{n =1} \nu _{n.} 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

\mu be σ-finite. Then there are measurable disjoint sets B _{1,} B _{2,} ... with \mu(B _{n)<} inf ty andc up ^{inf ty} _{n =1} B _{n=X} Then the measures\nu _{n( ⋅ ):=} \mu( ⋅ \cap B _{n)}

are finite and their sum is

\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\} with the σ-algebra l A = \{\{a\}, \emptyset\} . For all n \in \N , let \nu _{n} be the counting measure on this measurable space and define\mu: = \sum ^{inf ty} _{n =1} \nu _{n.} The measure \mu is by construction s-finite (since the counting measure is finite on a set with one element). But \mu is not σ-finite, since\mu(\{a\}) = \sum ^{inf ty} _{n =1} \nu _{n(\{a\})=} \sum ^{inf ty} _{n =1} 1= inf ty. So \mu cannot be σ-finite.

Equivalence to probability measures

For every s-finite measure

\mu = \sum ^{inf ty} _{n =1} \nu _{n} , there exists an equivalent probability measure P , meaning that \mu \sim P . One possible equivalent probability measure is given byP = \sum ^{inf ty} _{n =1} 2 ^{-n} (\nu _{n})/(\nu _{n(X)}) .

References

^[1]

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

Notes and References

1. Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling . 77 . Switzerland . Springer . 21. 10.1007/978-3-319-41598-7. 978-3-319-41596-3.