1U DR. J. MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



Now, if, is an arbitrarily assigned positive number, we can choose r, so small that 



Further, the measure of j, tends to zero when n increases indefinitely, in virtue of 

 hypothesis as to the relation between g (s, n) and g (*) ; hence 



our 



lim 



im f \f(s)\ds = 

 *-"J* 



We can, therefore, find a number N such that 



for all values of n 2: N. It follows that, for n 2: N, 



}b ft 



g (s, n)f(s) ds- g (*)/(*) ds 

 a Ja 



which establishes the theorem. 



2. A theorem corresponding to that of the preceding paragraph holds for 

 functions of two or more independent variables. The proof in each case follows the 

 same lines as that which has just been given. It will, therefore, be sufficient to state 

 the theorem for two independent variables : 



Let g (s, t, n) be a function defined at all points of the square, Q, which consists of 

 the points for ichich a^s^~b, a^t^b, and for a set of values of n which have + co 

 for an improper limiting point. Let the function be such that (1) the upper limit of 



\g(s,t,n)\ is a finite number g, and (2) g (s, t, n) (ds dt) exists as a Lebesgue 



J(Q) 



integral for each value of n. Let g (s, t, n) be related to a limited function g (s, t) 

 defined in Q in such a way that 



lim g (s, t,n} = g (s, t) 



n ^^ oo 



either at each point of Q, or at those points of it which do not belong to a certain set 

 of zero measure. Then, iff(s, t) is any function which possesses a Lebesgue integral 

 in Q, we have 



'' t} (ds df) = L g (s ' t} /(s> 4) (ds dt} - 



If /(s) possesses a Lebesgue integral in (a, b), f(s)f(t) possesses a Lebesgue 

 integral in Q. It follows that, with the hypothesis of the theorem just enunciated, 



= }^g(s,t)f(s)f(t)(d s dt). 



