212 Mr Dixon, On a property of summable functions. 



Then for any x which is common to §,.„,, and h',.^m and which is 

 not in r,. or TV, 



\f{x + e)-f{x)\<{U-L)l{n-l). 



Now the part common to Sy^,„ and h'r,m is K,m— 0, and of this 

 at most 2€r,m lies in F,. or TV- 



Hence \f(x + 6) -f(x) \<(U- L)l{n - 1) 



over intervals, E, which are together 



. n p 



7'=] m = l 

 P 



Also S X Br,m>'X^r — ne>{b — a)-n€, 



r m=\ 



P P 



The sum of the intervals E is therefore within 

 n (e + 2a) + np6 

 of the length of the whole interval (a, b). 



Again, \f(x + 0)—f(x)\ can nowhere exceed U—L. 



rb 

 Hence I {/(^ + ^)—/(a;)}^(ia; does not exceed ^ 



J a 



{^;^)\h-a) + (U-Ly{n{e + 2oc) + np0], 



the first term arising from E, and the second from the rest* of 

 (a, b). 



Here we may give n any fixed value, and put a=6= — , so 



fixing p ; then if ^ < — - , the whole will be < - of a quantitv 

 n-p n n J 



independent of n, and will tend to zero as n is increased. 



Hence Lt I {/(a; + ^)— /(ic)}2(^ic = 0, when/(a;) is any limited 

 e-9-o -' a 

 summable function. 



The same method shews that 



I \f(x+d)-f(x)\9dx 



* Here, as in § 4 below, such end-points of the intervals constituting e,., „ as 

 lie -within 5,.^ ^ form an enumerable set, whose measure is zero, so that it does 

 not matter whether they are included, excluded, or counted twice over. 



