CoNKAN — The Riemann Integral and Measurable Sets. 15 



JExample 3. Let U be the non-dense closed set complementary to the set 

 of intervals [a] described in the preceding 

 example. Then the plane set S of points [x,y], 

 where x is any point of U and < _?/ < 1, is 

 closed. Let us determine the double integral of 

 scy in the set *S. 



The internal points of the unit-square which 

 are not points of S form an open domain of the 

 second kind D. We have to find 



fl."// . (le. 



Fio. 3. 



M 



The repeated integral 



dx xy dy = \ content of \a] 



is found to exist, and is therefore by A equal to the double integral. 

 Finally, 



xyde=\ dx\ xy dy - xy de - \ - \y. content of \a\ =\J, 



where J is the plane content of ;S. 



In this example, we have an instance of the possibility referred to in 

 § 14, C. — that the region of the 2nd integration in the repeated integi-al 

 may not be extensible to the whole interval < a; < 1. 



