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2015/06/10

measures 10

On an arbitrary space \Omega , if there is an outer measure m^o:a\subset\Omega \rightarrow \mathbb{R} , the collection of sets
\mathcal{F}=\left\{a\subset\Omega|\forall e\subset\Omega,m^o(e)=m^o(e\cap a)+m^o(e\cap a^c) \right\}
could be defined.

It has been proved that for the elements in this collection \mathcal{F} ,
if a_1,a_2\in\mathcal{F},a_1\cap a_2=\phi , then (a_1\cup a_2)\in\mathcal{F} .

Although the proof is not easy, the number of set in the collection can be extended from 2 to \infty . Namely, for a_1,a_2,\cdots  and a_i\cap a_j=\phi , then (\cup a_i)\in \mathcal{F} .

The collection \mathcal{F} is called \sigma algebra, and
on \sigma algebra, an outer measure m^o  becomes a measure m .








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