FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 113 



sequences), and, apart from their applications, have no connection with the theory 

 developed below. It will, therefore, be convenient to enunciate and prove them at 

 the outset. 



The first theorem is : 



Let g(s, n) be a function defined for nil values of s in the interval (a, b), and for a 



set of values ofn which have + 00 for an improper limiting point.. Let this function 



fi 

 g (s, n) ds 

 a 



<:rixts as a Lebesgue integral for each value ofn. Let g(s, n) be related to a limited 

 function, g (s), defined in (a, b) in such a way that 



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



H -> 00 



either at each point of (a, b) or at those points of (a, b) which do not belong to a 

 certain set of zero measure. Then, if f(s) is any function which possesses a Lebesgue 

 integral in (a, b), we have 



Jim f * g (,, )/(,) ds = f 'g (.)/(*) ds. 



J a * 



To prove this, let us take any positive number y. Let us denote by j\ the set of 



points of (a, b) at which 



\g(s,n)-g()\>r)* 



and by J, the set complementary to _;',,. Then we have 



f* 9 (. )/() ds ~ f 9 ()/() d * = [ + f (* u )-!/ (*)]/ 



Jo Jo *;. * 



The numerical value of the first integral is evidently not greater than 



where ^ is the upper limit of | g (s) \ in (a, 6), and the numerical value of the second 

 is not greater than 



which is not greater than 



ifl/MI* 



J a 



We thus have the inequality 



\{'g(*,n)f(s)ds-\ k g(s)f(s)ds 



I J a Jo 



* It should be observed that, in virtue of its relation to the functions g (t, n), g(*) is summable, and that 

 j n is therefore measurable. 



VOL. CCXI. A. Q 



