# why we use σ-Field in probability theory?

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?

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