CoNRAN — The Riemann Integral and Measurable Sets. 11 



In the investigation given below it will be seen that there are four 

 inequalities, two of which refer to the case when the y-integration is taken 

 over the projection of the domain on the ^/-axis ; the remaining two refer to 

 the case when the ^-integration is taken over the side of the fundamental 

 rectangle. The ^.'-integration is in each case taken over a section of the 

 domain parallel to the a;-axis. 



Let f{x, y) be finitely bounded in an open domain D of the second kind 

 contained in a fundamental rectangle whose sides are parallel to the axes of 

 coordinates. Let F denote the projection of the domain on the ;/-axis ; this 

 projection consists of a set of non-overlapping open intervals. Let Gy denote 

 the section of the domain at an ordinal distance y from the s-axis. Gy con- 

 sists also of a set of open intervals, and the upper integral of f[o:y) over the 

 section G,j may be determined. Denoting this integral by J^y) we can find 

 the lower integral of J[y) over the set F. The result will be shown to be 



< I f.de. 



Suppose the plane divided into elementary squares by parallels to the 

 axes, and let R denote the rectilinear area formed by those squares which 

 are internal to D. Those points of D which are external to R form an open 

 domain Di. We may suppose that the content of Di is < e, where e is any 

 assigned positive quantity. Then 



, I = 1 1 U ^1/e, 



D J -K I \ J Oi] 

 where M is the maximum of | /(.r, //) | . 



Now, let a second system of parallels, including the first, be taken, 

 dividing the squares into elementary rectangles. Let the area of the 

 rectangle whose sides are x = x,-, cc = Xr^u y - Vs, y = Vs+i, be denoted 

 by a,-5, and let M,-s be the upper limit of /(», y) in an, and form the sum 

 2Jir»«rj, extending to all the rectangles contained in R. The second 

 system of parallels can be determined so that 



B 



r 



Hence, - 2J/,,.a„ < t + -l/.c 



I J^ I 



Now, let y,<y < y,^i, therefore, 



f f(,cy) dx < ^Mr,(xr,, - .» + M.I (y), 

 where I{y) is the content of the section of the domain B^. 



[2»] 



