ABSTRACT

T h e o r e m 5.6. Let μ η be σ -additive measures on (N, ?(N)) taking only the values 0 and 1. Suppose μ{Ε) = lim ^oo μ η{Ε) exists for all EC. N. Then μ is a σ -additive measure.

P r o o f . If μ is not σ-additive then μ = бң for some H e *N \ N. Let A = {N e N : 3n e N μ η = <5дг}; for every n there is an N e A with μ η =δχ . Thus μ„(Α) = 1 for every n, so μ (A) = 1. As H e * A and H is infinite, A is infinite, so A can be partitioned into two disjoint infinite sets C and D. H is in either *C or *D but not both; for definiteness, assume H $ *D. Then 0 = p(D); but also μ(Ο) = lim ^oo μ η(£>) = limneD μ η(ΰ) = 1, a contradiction. Ч

5.2. Measures on N. I now generalize the above to arbitrary finite measures (not just point masses) on N. The analogue of Proposition 5.4 is the following.