FUNCTIONS IN THE THEOKY OF INTEGRAL EQUATIONS. 125 



K (s, t) satisfy the requirement of the theorem of the preceding paragraph. Then, for 

 any fixed value of $ belonging to (a, b), we have 



It is easily proved from this that the numbers 



are everywhere dense in the interval (L(s), U(s)).* 



We can therefore find a sequence of these numbers which has any given number 

 belonging to the closed interval (L (s), U (s)) for its limit. Keferring to the theorem 

 of 4, we see that : 



If, in addition to the hypotheses of the theorem of 4, it is assumed that the upper 

 limit of | \ji n (s) \ in (a, b) is less than a fixed number for all values of n, then, by the 

 introduction of suitable brackets, the series 



may be made to converge to either of the limit* 



provided that this limit is finite. 



8. Returning to the system of normal functions of 5, let us suppose it to be 

 such that 



for each limited function <f> (s) which has a Lebesgue integral in (a, b). By considering 

 the function which is identical with <f> (s) in an interval (a^ b^ of (a, b) and is zero 

 elsewhere, we see that 



H = 1 



provided that <f>(s) is limited and has a Lebesgue integral in (a lt />,). 



Let us now suppose f(s) to be a function, defined for all values of s, which has a 

 Lebesgue integral in any finite interval. Let x (* be a limited function which has 

 a Lebesgue integral with respect to t in (a,, &i), for each value of * in a certain closed 

 interval (a a> 1 3 ) ; further, let us suppose that x (*, is a uniformly continuous function 

 of s in (03, & 2 ), for values of t belonging to ( 6,). In virtue of the latter condition, 



* Of. HOB.SON, 'The Theory of Functions of a Real Variable,' p. 712. 



