Course in Probability Theory

382 1 SUPPLEMENT : MEASURE AND INTEGRALBoth these collections belong to ~T because the Borel field is closed undercountable union and intersection, and these operations may be iterated, heretwice only, for each collection . If B E . ~b , then B belongs to both %os andz~bba because we can take B,,,,, = B . Finally, A E Avb if and only if A` E As,becausen /n U Bmn = U B mnIn nm nTheorem 4 .Let A E T * . There exists B E &b such thatA CB ; It* (A) = A* (B) .If u is a-finite on ~, then there exists C E Aba such thatC C A ; ,u*(C) = A*(A) .PROOF . For each m, there exists {Bmn } in ~T such thatPutA C U BmnnnB,n = U Bmn ;nA * (Bmn) :!~ A * (A) + 1 .mB = n Bmmthen A C B and B E A~b .We haveh * (B) < p * (Bm)