Course in Probability Theory

4 INTEGRAL 1 401Theorem 9 . Let If, } be an increasing sequence of functions in ~T- + withlimit f : f n T f . Then we haveThen we havelim T E(f n) = E(f) < + oo .nPROOF . We have f E ~+ ; hence by Definition 8(b), (41) holds . For eachf,, we have, using analogous notation :(42) lim f E(fnm) ) = E(f n ) .MSince f n T f , the numbers )2m f n (a))] T)2' f (w)] as n t oo, owing to theleft continuity of x -p )x] . Hence by Corollary to Theorem 8,(43) urn f E(fnm) ) = E(f (m) ) .It follows thatlmm T 1nm T E(f nm) ) = lmm f E(f (m)) = E(f ) .On the other hand, it follows from (42) thatlim f lim f E(f nm)) = lim t E(f n ) .n m nTherefore the theorem is proved by the double limit lemma .From Theorem 9 we derive Lebesgue's theorem in its pristine positiveguise .Theorem 10 . Let f n E F+ , n E N . Suppose(a) lim n f n = 0 ;(b) E(SUp n f n ) < 00 .(44)PROOF .Put for n E N :lim E (f n ) = 0 -n(45)gn = Sup f kk>nThen gn E ~T+ , and as n t oc, g n ,. lim sup„ f n = 0 by (a) ; and g i = sup,, f „so that E(gl) < oc by (b) .Now consider the sequence {g l - g n }, n E N . This is increasing with limitg i . Hence by Theorem 9, we havelim T E(gi - gn) = E(gl ) .n