252 Dlt. W. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 



rectangle ; this additiou of the rectangle is therefore equivalent to sinking the axis 

 of x to a convenient position, so that all the blocks become positive. The upper 

 integral of the given function, plus the constant, is now represented by the set got 



by closing the set of blocks. From this we have now 

 to subtract again the rectangle, or, which is the same 

 thing, return the a/'-axis to its original position, iu 

 order to get the geometrical representation of the 

 upper integral of the given function. 



~Fie~2~ Thus, in the general case, the geometrical repre- 



sentation of the upper integral is not precisely a 



closed set, but a closed set minus a rectangle, as is shown roughly iu fig. 2, viz., the 

 shaded region, of which the part below the a;-axis, corresponding to E 2 , is to be 

 considered as negative, and the other part, corresponding to EI, is closed. 



Similarly, in the case of a function which is always negative, the geometrical 

 representation of the lower integral is a closed plane set constituted by blocks ; in 

 the general case it is represented by the excess of a rectangle over such a set. 



When the fundamental set S is any measurable set whatever, instead of closing 

 the fundamental set actually, we do so relative to S, that is, we take in only those 

 limiting points which lie on ordinates through S. The rectangle to be subtracted in 

 the case of the ordinary upper integral, or from which the relatively closed set is to 

 be subtracted in the case of a lower integral, is then a relative rectangle, that is, that 

 part of a rectangle which lies on the ordinates through S. 



