Suppose G,G, ..., Gn are disjoint subsets of such that 72 n = UG. i=1 Prove that the family G consisting of o-field on 2. and all unions of some (or all) of G, ..., Gn constitutes a b) Prove that any finite o-field F on is of the type described in a). c) Let F be a finite o-field on 2 and let Y: R be F-measurable (o-field generated by Y satisfies o(Y) CF). Prove that Y assumes only finitely many possible values. More precisely, there exists a disjoint family of subsets F,..., Fm E F and real numbers ,9m such that m Y(w) = yil F, (w). i=1