Course in Probability Theory

22 1 MEASURE THEORYAxiom (ii) is called "countable additivity" ; the corresponding axiomrestricted to a finite collection {E j ) is called "finite additivity" .The following proposition(1) En ., 0 =#- ) - ~- 0is called the "axiom of continuity" . It is a particular case of the monotoneproperty (x) above, which may be deduced from it or proved in the same wayas indicated below .Theorem 2 .2 .1 . The axioms of finite additivity and of continuity togetherare equivalent to the axiom of countable additivity .If E, 4,PROOF . Let E„ ~ . We have the obvious identity :00 00E n = U (Ek\Ek+1) U n Ek .k=n k=10, the last term is the empty set . Hence if (ii) is assumed, we haveVn > 1 : J (E n ) _00k=n-,~)"(Ek\Ek+1)~the series being convergent, we have limn,,,,-,"P(En) = 0 . Hence (1) is true .Conversely, let {Ek , k > 1) be pairwise disjoint, then00U E k ~ok=n+1(why?) and consequently, if (1) is true, then00lim :~, U Ek = 0 .)t -4 00(k=n+1Now if finite additivity is assumed, we haveU Ek = ?P U Ek + JP U EkC, ~'\k=1 k=1 k=n+1 =Ono+11 00?11(Ek) + ~) U Ek -k=1 k=n+1This shows that the infinite series Ek°1 J'(Ek) converges as it is bounded bythe first member above . Letting n -+ oo, we obtain