of an infinite series over an infinite range 13 



measure zero, because if it did not the above limit would be 

 infinite ; and so, for our purposes 



J a J b 



exists. 



Our convention has enabled us to infer the existence of the 

 inner integrals from the existence of the double limit. 



Note 3. A thorough treatment on different lines of the subject 

 of this paper will be found in two papers by Prof W. H. Young : 



(1) " On the change of order of integration in an improper 

 repeated integral," Trans. Gamh. Phil. Soc, xxi. p. 361. 



(2) "The application of expansions to definite integrals," 

 Proc. Lond. Math. Soc, ix. (1910), p. 463. 



In this paper we content ourselves with giving simple 

 generalisations of well-known results with proofs depending on 

 comparatively elementary theorems. There is no attempt to 

 obtain comprehensive results. 



