PROCEEDINGS 



OF THE 



Cambriirgc ISljtlasapIjkal Bomi^. 



On the term by term integration of an infinite series over an 

 infinite range and the inversion of the order of integration in 

 repeated infinite integrals. By S. Pollard, M.A., Trinity College, 

 Cambridge. (Communicated by Prof. G. H. Hardy.) 



[Received 1 January, 1920. Read 8 March, 1920.] 



The problem for infinite series. 



1. The problem to be solved is that of determining conditions, 

 under which the equation 



00 /•=» /"» 00 



2 Un(oc)dx==l S Un(oi;)dx, (1) 



n=l J a J an=l 



is true. It is discussed in detail in Bromwich's Infinite Series, 

 pp. 452-455, where various conditions are given. All these con- 

 ditions will be found to involve uniform convergence, the fact 

 being that the infinite integrals there considered are obtained as 

 limits of Riemann integrals and, in the theory of the latter, con- 

 siderations as to the validity of the equation 

 rb m rb oo 



lim I 2 M„ (x) dx = j S Uji (w) dx, (2) 



almost always involve uniform convergence. Thus conditions for 

 term by term integration over an infinite range, being built up 

 from the conditions for term by term integration over a finite 

 range, involve uniform convergence. 



Now the condition of uniform convergence is by no means a 

 necessary one : it occurs because of the lack of power in the 

 methods of the Riemann theory. Much wider conditions can be 

 obtained by the use of the Lebesgue theory. It is the object of 

 this paper to give these. 



VOL. XX. PART L 1 



