FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 115 



3. The second theorem, to which reference was made above, is : 

 Let g(s, t, n) he a function defined at all -point* of the rectangle, R, which cotisists 

 of the points for which a, < s s /;,, <t S t s b, and for a set of mines ofn which have 

 + oo for an improper limiting point. Let this function be such that (I) the upper 



f* 

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



Ja 



integral for all values of s and n. Let g (s, t, n) be related to a limited function 

 g(s, t) defined in R in such a way that, as n tends to oo, g(s, t, n) converges uniformly 

 to g(s, t) either in the u-hole of R, or, for each positive number tj, in those parts of it 

 which correspond to 



| --&, | 17, (m = 1,2?..., M), 



where the numbers ,, a z , ..., M are. all finite. Then, if f(t) is any function which 

 has a Lebesgue integral in (a, b), 



\'ff(,t,*}f(t)dt 



Ja 



converges uniformly to 



for a :5 c s- >, i s :5 &i, as n tends to oo. 



In the first place, let us assume that g (s, t, n) converges uniformly to g (s, t) in the 

 whole of R. When any positive number e is assigned, we can then choose N great 

 enough to -ensure that at each point of R 



\g(s,t,n)-g(,t)\ <J\\f(t)\dt 

 for all values of n > N. From this we see at once that 



\\[ff (, t, 



- g (*, /(>) dt 



that is to say, 



<, 



for a s c s 6, a l S s S 61, and n > N. The theorem is thus proved for this case. 



More generally, let us suppose that, for each positive number >?, g (s, t, n) converges 

 uniformly to g (s, t) in those parts of R which correspond to 



iJ-o^-A.laif, (m= 1, 2, ..., M); (1) 



then, selecting any value of 77, for each pair of values of c and * the points of 

 a < t S c which do not satisfy all the inequalities just written lie in a set (j til ) 



Q 2 



