of an infinite series over an infinite range 3 



For if lim f" ^ \f{l v)\d^dr) 



exists and is finite, then S^^y satisfies the condition of (7). We 

 have in fact 



^x, y — ^ x,y ^ x,yj 

 fx ry 

 where S'x,y= I \f{^,v)\d^dr}, 



J aJb 



S\y=r ("[\f{^,V)\-f{lv)]d^dv, 

 J aJ b 



and both S'cc,y, 8"x,y are increasing in x and y and have a finite 

 double limit — the former by hypothesis and the latter because 



Note. The above results still hold when either or both of the 

 variables x, y take only positive integral values. 



Definition of infinite integrals. 



4. Let /(a;) be any function which is summable in (a, X) for 

 all X greater than a. 



If lim f{x) dx, 



X-*-ix J a 



where the integral is taken in the sense of Lebesgue, exists and 

 is finite, we say that 



J 



f (x) dx 



converges and attribute to it the value of the limit. 



This definition is evidently consistent with and more general 

 than that usually given, where /(a:;) is assumed to be integrable in 

 Riemann's sense in (a, X). It has the special advantage of not 

 being restricted to functions which are bounded in every (a, X). 

 And we lose nothing by adopting it, as the two theorems on which 

 the theory of infinite integrals rests, the first and second mean 

 value theorems, are still true when we abandon the restriction 

 that f{x) is to have a Riemann integral and make only the 

 assumption that /(a;) is summable*. 



General theorems. 



rX m 



rX m 

 5. I. If the double limit lim I 2 m„ (x) dx exists and 



m-^-oo, X-*-<x>J an=l 



(a) I Un (x) dx converges for all n, 

 J a 



Ibid. t. II. 2nd Ed., p. 53. 



