s-finite measure

Let $(X,\; \backslash mathcal\; A\; )$ be a measurable space and $\backslash mu$ a measure on this measurable space. The measure $\backslash mu$ is called an s-finite measure, if it can be written as a countable sum of finite measures $\backslash nu\_n$ ($n\; \backslash in\; \backslash N$),

:$\backslash mu=\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash nu\_n.$

The Lebesgue measure $\backslash lambda$ is an s-finite measure. For this, set

:$B\_n=\; (-n,-n+1]\; \backslash cup\; [n-1,n)$

and define the measures $\backslash nu\_n$ by

:$\backslash nu\_n(A)=\; \backslash lambda(A\; \backslash cap\; B\_n)$

for all measurable sets $A$. These measures are finite, since $\backslash nu\_n(A)\; \backslash leq\; \backslash nu\_n(B\_n)=2$ for all measurable sets $A$, and by construction satisfy

:$\backslash lambda\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash nu\_n.$

Therefore the Lebesgue measure is s-finite.

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 $\backslash mu$ be σ-finite. Then there are measurable disjoint sets $B\_1,\; B\_2,\; \backslash dots$ with and

:$\backslash bigcup\_\{n=1\}^\backslash infty\; B\_n=X$

Then the measures

:$\backslash nu\_n(\backslash cdot):=\; \backslash mu(\backslash cdot\; \backslash cap\; B\_n)$

are finite and their sum is $\backslash mu$. 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 $X=\backslash \{a\backslash \}$ with the σ-algebra $\backslash mathcal\; A=\; \backslash \{\backslash \{a\backslash \},\; \backslash emptyset\backslash \}$. For all $n\; \backslash in\; \backslash N$, let $\backslash nu\_n$ be the counting measure on this measurable space and define

:$\backslash mu:=\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash nu\_n.$

The measure $\backslash mu$ is by construction s-finite (since the counting measure is finite on a set with one element). But $\backslash mu$ is not σ-finite, since

:$\backslash mu(\backslash \{a\backslash \})=\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash nu\_n(\backslash \{a\backslash \})=\; \backslash sum\_\{n=1\}^\backslash infty\; 1=\; \backslash infty.$

So $\backslash mu$ cannot be σ-finite.

For every s-finite measure $\backslash mu\; =\backslash sum\_\{n=1\}^\backslash infty\; \backslash nu\_n$, there exists an equivalent probability measure $P$, meaning that $\backslash mu\; \backslash sim\; P$. One possible equivalent probability measure is given by

:$P=\; \backslash sum\_\{n=1\}^\backslash infty\; 2^\{-n\}\; \backslash frac\{\backslash nu\_n\}\{\backslash nu\_n(X)\}.$

{{cite book |last1=Kallenberg |first1=Olav |author-link1=Olav Kallenberg |year=2017 |title=Random Measures, Theory and Applications|series=Probability Theory and Stochastic Modelling |volume=77 |location= Switzerland |publisher=Springer |page=21|doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3}}

• {{cite journal|last1=Falkner|first1=Neil|title=Reviews|journal=American Mathematical Monthly|volume=116|issue=7|year=2009|pages=657–664|issn=0002-9890|doi=10.4169/193009709X458654}}
• {{cite book|author=Olav Kallenberg|title=Random Measures, Theory and Applications|url=https://books.google.com/books?id=i6WoDgAAQBAJ|date=12 April 2017|publisher=Springer|isbn=978-3-319-41598-7}}
• {{cite book|author1=Günter Last|author2=Mathew Penrose|title=Lectures on the Poisson Process|url=https://books.google.com/books?id=JRs3DwAAQBAJ|date=26 October 2017|publisher=Cambridge University Press|isbn=978-1-107-08801-6}}
• {{cite book|author=R.K. Getoor|title=Excessive Measures|url=https://books.google.com/books?id=UxvSBwAAQBAJ&pg=PA182|date=6 December 2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3470-8}}

{{Measure theory}}

Category:Measures (measure theory)