Course in Probability Theory

2 CHARACTERIZATION OF EXTENSIONS 1 383Since St n E ~ and Bn E aQ , it is easy to verify that S2„Bn E .7u s , by thedistributive law for the intersection with a union . PutIt is trivial that C E % SQ andConsequently, we haveC = U Q n B' nnnA = U S2 1A D C .nµ * (A) > µ*(C) > Jiminf/c * (S2 nBn)n= lim inf µ* (Q,,A) -,a* (A),nthe last equation owing to property (e) of the measure A* . Thus A* (A) =A* (C), and the assertion is proved .The measure A* on 3T* is constructed from the measure A on the field. The restriction of [c* to the minimal Borel field ~ containing ~?'b7- willhenceforth be denoted by A instead of µ* .In a general measure space (Q, v), let us denote by ..4"(fi', v) the classof all sets A in ~ with v(A) = 0 . They are called the null sets when -,6' andv are understood, or v-null sets when ry is understood . Beware that if A C Band B is a null set, it does not follow that A is a null set because A may notbe in ! This remark introduces the following definition .DEFINITION 7 . The measure space (Q, ~', v) is called complete iff anysubset of a null set is a null set .Theorem 5 . The following three collections of subsets of Q are idential :(i) A C S2 and the outer measure /.c * (A) = 0 ;(ii) A E ~T * and µ*(A) = 0 ;(iii) A C B where B E :% and µ(B) = 0 .It is the collection ( "( *, W) .PROOF . If µ* (A) = 0, we will prove A E :?X-7* by verifying the criterion(3) . For any Z C Q, we have by properties (a) and (b) of µ* :0 < µ*(ZA) < µ* (A) = 0 ; u * (ZA`) < /.t* (Z) ;