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i read that sigma field is used to make a measurable space .

but

why don’t we define the measure on the whole omega (sample space) (set of all possible out comes) ?

Wikipedia says “In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra”

what other kinds of useful measures do we have in probability ? ( i mean non_theoretical ones ) that we cannot have if we define the measure over the whole omega?

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The Lebesgue measure on $[0,1]$ is an important example for a (probability) measure which cannot be defined on the power set $\mathcal{P}([0,1])$.

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  • $\begingroup$ could you explain more. what is Lebesgue measure and why it cannot be defined on the power set. $\endgroup$ – kiyarash Dec 28 '14 at 17:58
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    $\begingroup$ If you aren't familiar with Lebesgue measure and integration then you should probably study a bit of that before trying to dive into measure-theoretic probability. Terence Tao has a free introductory text on his website: terrytao.files.wordpress.com/2011/01/measure-book1.pdf $\endgroup$ – Math1000 Dec 28 '14 at 22:42

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