8 Mr Pollard, On the term by term integration 



exists and is finite. Then lim (lim ^«t,x) exists and is finite, 

 and we get a contradiction. And if 



^ \Un{x)\dx 

 J an=l 



exists and is finite, then so does 



rx m 

 \ 2 I ti„ {x) I dx 



J a 7i=l 



for all X greater than a. Hence as in theorem II 

 rX oa rX TO 



I "E \ Un {w) \ dx = lim X I Wn (^) I ^^> 



J a 71=1 Wi-»-oo J a 71 = 1 



and it follows that 



r"" 00 



E \un\x)\dx= lim (lim S^^x), 



Ja n = l X-*-oo 7j,-*-oo 



and we again get a contradiction. 



Thus the first case alone is possible, and this is the case in 

 which our theorem is true. 



C. Write r u,{w)dx = g,{X), 



- « 



m 

 tgn{X)^Sm,X. 



71=1 



00 



Since S g^ (X) converges uniformly for a ^ X, given e > we 



71=1 



can find No such that 



I i gAX)\<^, {X>a,N^N-,). 



n=N+l 



Thus \Sm,x- i 9n (X) I < e, {X^a, m ^ N^). 



71 = 1 



00 



Hence if lim S gn{^) exists and is finite, so does lim aS'^^^ x , 



X-*-^ n=l 7?l-»Q0 , JC-».Xi 



and the two are equal. Now 



I i: g^ (ZO -i g, (Z") I ^ i f g, (Z) - % g, (Z") j 



tt=l 71=1 71=1 n=l 



+ 1 i gn{X')\ + \ i 5'n(X")I 

 n=N+l n=N+l 



^1 f 5r,(Z')-i5'«(X")l + 26. 



71=1 71=1 



But since lim g^ (Z) (= u^ (w) dx) exists and is finite for 



X-*oo J a 



