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