Suppose G,G, ..., Gn are disjoint subsets of such that 72 n = UG. i=1 Prove...

  

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Suppose G₁,G₂, ..., Gn are disjoint subsets of such that 72 n = UG₁. i=1 Prove that the family G consisting of o-field on 2. and all unions of some (or all) of G₁, ..., 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 F₁,..., Fm E F and real numbers ₁,9m such that m Y(w) = yil F, (w). i=1 Suppose G₁,G₂, ..., Gn are disjoint subsets of such that 72 n = UG₁. i=1 Prove that the family G consisting of o-field on 2. and all unions of some (or all) of G₁, ..., 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 F₁,..., Fm E F and real numbers ₁,9m such that m Y(w) = yil F, (w). i=1