o/ an infinite series over an infinite range 9 



N 



each n, so does lim S gn{^) ^-nd we can find X^ such that 

 1 f 5r„ (Z') -I g, {X") 1 < 6. {X', X" ^ Zo). 



71=1 »=1 



Hence 1 I ^„ (X) - 2 ^„ (Z") | < 3e, (Z', Z" ^ Z„) 

 and therefore, by the general principle of convergence 



00 



lim 2 gr, {X) 



X-»-t» 71=1 



exists. Thus lim *S^„i,x exists. The other conditions of (I) are 

 satisfied by hypothesis and our theorem follows. 



The problem for infinite integrals. 

 7. We have to determine conditions under which the equation 



dx\ f{x,y)dy==\ dy \ f{x,y)dx (5) 



J a J b J b J a 



is true. The methods adopted above apply almost without change 

 and we get conditions almost identical with those already given. 

 We quote them without "proof, as the proofs can be made up 

 immediately on the lines of those already given. 



As regards the nature oif{x, y), we assume throughout that 

 f {x, y) is summable in the region 



{a^x^X, b^y^Y), 

 for all Z ^ a, Y ^b ; so that, by Fubini's theorem*, the repeated 



rX rY fY rX 



integrals dx \ f{x,y)dy, I dy \ f{x,y)dx exist and are 



J a ■ b J b J a 



equal to the double integral. 



General theorems. 



rX rY 



and 



8. I'. If the double limit lim I j f{x, y) dxdy exists 



(a) I f{x,y)dx, converges for y^b, * 



(b) I f{x,y)dy, converges for x '^ a, 



rX rY rX r^ 



(c) lim dx f{x, y) dy, dx f{x, y) dy, 



F-w-oo J a J b J a J b 



* De la Valine Poussin, Integrales de Lebesgue etc., p. 53. 



