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

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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

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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

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Relation to σ-finite measures

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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

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For every s-finite measure  , there exists an equivalent probability measure  , meaning that  .[1] One possible equivalent probability measure is given by

 

References

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  1. ^ a b Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 21. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.