12 Mr Pollard, On the term by term integration 



Now in the Lebesgue theory the integral of any summable 

 function over a set of zero measure is zero, and consequently we 

 may neglect a set of measure zero without affecting the value of _ 

 the integral. Hence when we are faced with the problem of find- l{ 

 ing the value of a function which is indefinite or infinite at the 

 points of a set of measure zero, we simply neglect these points 

 and find the value of the integral over the residue. This is taken 

 to be the value of the integral over the original set. 



With the above convention it is true that, if 



I /(^, y) dxdy 

 Ja Jb 



exists, so does dy f{x, y) dxdy, 



. h J a 



although there may be points in (6, Y) at which the single 



integral 



rx 

 f{x,y)dx 



f 



J a 



does not exist. 



It is always to be understood in dealing with repeated 

 Lebesgue integrals (finite or infinite) that the inner integrals 

 need only exist at all the points of the range of integration of the 

 outer integral save those of a set of measure zero. 



Let us apply the foregoing remarks to theorem B'. Suppose 



rY rx 



dy \ \f{^,y)\dx 



■lb J a 



exists. Then we know that 



lim 11 fix, y) I dxdy 



X-*-x , F-9.20 J a J b 



exists. It follows that 



I dx\ \f{x,y)\dy, 



J a Jb 



on of Y, is bounded £ 

 lim / dx \f{x,y)\dy 



T-^x J a J b 



considered as a function of Y, is bounded as F tends to infinity. 

 Thus 



rx rY 



Y- 

 exists and is finite. It follows that 



\f{x, y) \ dy 



b 



converges at all the points of (a, X) save possibly those of a set of 



