Course in Probability Theory

376 1 SUPPLEMENT : MEASURE AND INTEGRAL(d) (additivity) if {B J }UJ BJ E :%, thenis a countable sequence of disjoint sets in tand(1)It(UBi) -i )Jg(Bj) .Let us show that the properties (b) and (c) for outer measure hold for a measureg, provided all the sets involved belong to A .If A 1 A2 E z~b, and A 1 C A 2 , then A 'A2 E A because ~5t- is a field ;A 2 =A 1 UA~A 2 and so by (d) :g(A2) = /,t(A1)+ g(A'A2) ? A00-Next, if each AJ E ~b , and furthermore if U J A J E A (this must be assumedfor a countably infinite union because it is not implied by the definition of afield!), thenJ-A 1 UAlA2 UAlA2A 3 U . . .and so by (d), since each member of the disjoint union above belongs to : 3 :g UAJ = A(Ai) + g(A,A2) + A(AlA2A3) + . . .I< g(A1)+g(A2)+A03)+ . . .by property (b) just proved .The symbol N denotes the sequence of natural numbers (strictly positiveintegers) ; when used as index set, it will frequently be omitted as understood .For instance, the index j used above ranges over N or a finite segment of N .Now let us suppose that the field : b is a Borel field to be denoted byand that g is a measure on it . Then if A„ E '- for each n E N, the countableunion U, A„ and countable intersection nn A„ both belong to ~T- . In this casewe have the following fundamental properties .(e) (increasing limit) if A„ C A, + 1 for all n and A, T A - Un A,,, thenlim T g(A„) = A(A) .17(f) (decreasing limit) if A„ D A„ + 1 for all n, A„ 4, A = nn A,,, and forsome n we have g (A„) < oo, then1im ~ A(An) = A(A) .11