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Iii) Finally if μ is finite (or σ-finite) on P, then ν is finite (or σ-finite) on R. Proof (i) Suppose that μ is finitely additive on P, and let E ∈ R. 3, E = ∪n1 Ej where the Ej are disjoint sets of P. Define ν(E) = n1 μ(Ej ). We must check that ν is well defined. That is, if E can also be written as ∪m1 Fk , for disjoint sets Fk ∈ P, it must be verified that mk=1 μ(Fk ) = n j=1 μ(Ej ). To see this, write Hjk = Ej ∩ Fk . The Hjk are all disjoint sets of P and ∪mk=1 Hjk = ∪mk=1 (Ej ∩ Fk ) = Ej ∩ E = Ej , whereas similarly ∪nj=1 Hjk = Fk .

This holds whatever measure μ is on R. e. whether μ is the only measure on S(R) such that μ(E) = μ(E) when E ∈ R. 7 that this is the case if μ is σ-finite on R. This is shown, and the results thus far summarized, in the following theorem. 3 (Caratheodory Extension Theorem) Let μ be a measure on a ring R. e. μ(E) = μ(E) if E ∈ R). If μ is σ-finite on R, μ is then the unique such extension of μ to S(R), and is itself σ-finite on S(R). Proof The existence of μ has just been shown. e. μ1 (E) = μ(E) = μ(E) for all E ∈ R).

4 n i=1 f (xi ), μ(∅) = 0. Show that μ is a measure on the ring R. (If f (x) ≡ 1, μ is called counting measure on R. ) Let E be a class of sets and μ be a measure on R(E) such that μ(E) < ∞ for all E ∈ E. Show that μ is a finite measure on R(E). Let X be the set {1, 2, 3, 4, 5} and let P be the class of sets ∅, X, {1}, {2, 3}, {1, 2, 3}, {4, 5}. Show that P is a semiring. Define μ on P by the values (in the order of the sets given) 0, 3, 1, 1, 2, 1. Show that μ is finitely additive on P. What is R(P)?

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