o/ an infinite series over an infinite range 1 1 



D'. If \r fix, y)dy\^^x{x) for a^ x^ X, 6 ^ F, 



J b 



ivhere yjrx is summahle in (a, X), and 



\ dy \ f {x, y) dx 

 Jb J a 



converges uniforndy for a ^ X, X being arbitrary, and 



f{x, y) dx 



converges for y^b; then both sides of {b) exist and are finite and 

 equal. 



10. Note 1. Results B are especially valuable, as they are 

 easy to remember and convenient to apply. The power of the 

 Lebesgue theory is shewn very clearly here in that by using it 

 we are enabled to make the hypothesis which ensures the exist- 

 ence of the double limit* ensure also the passage to the limit under 

 the sign. 



Note 2. It is well to be precise as to the meaning of the word 

 "exists" as used in connection with repeated Lebesgue integrals. 



Suppose /(^, y) is measurable in x, y in the rectangle 



^a^x^X\ 

 Kb^y^Yj' 



We know that the function y (a:;, y) considered as a function of 

 X, is measurable in (a, X) for each y in (6, Y) a set of zero measure 

 being excepted. It may not, however, be summable in (a, X), i.e. 



f{x,y)dx 



may not exist, for all ?/ concerned. But, if/(^, y) is summable 

 over the rectangle, i.e., if the double integral 



rx rY 



fix, y)dxdy 



■ b 



exists ; then it can be shewn that 



rx 

 fix, y)dx 



f 



J a 



exists for all values of y in (b, Y), save possibly those of a set of 

 measure zero. 



CO ,-co 



* The existence of 2 I j u^^ (x) \ dx implies the existence of the double limit 

 n=lj a 



by (7) of §3 ; and in addition, by the use of (C 1) on | u„(x) | , it will be found to 

 imply the validity of the passage to the limit under the sign. 



