24(J DK. W. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION 7 . 



27. It is instructive to show the identity of LEBESGUE'S geometrical definition 

 with that of 24 (S being a finite segment) directly. At the same time it should be 

 remarked that the argument is independent of the fact that S is a finite segment, so 

 that we have some general theorems with respect to the geometrical representation of 

 the processes of integration with respect to a set of points to which I shall return 

 in 32. 



Lemma 1. If at every point of a measurable set G on the x-axis of content I ive 

 place an ordinate of length p, the plane set constituted by this set of ordinates, or, 

 as I shall call them, for definiteness, thin set of block*, is- measurable and has 

 content pi. 



For we can enclose G in a set of non-overlapping intervals of content less than 

 I + e. Erecting on these rectangles of height p + e, we have enclosed the whole plane 

 set in a set of rectangles of content as near as we please to pi. On the other hand, 

 taking in E a closed set of content greater than I e, the corresponding set of 

 blocks forms a closed set of content as near as we please to pi. Thus the outer 

 measure of the content of the set of blocks is not greater than pi, and the inner 

 measure of the content is not less than pi, which proves the Lemma. 



Suppose now we are given any measurable set S, and on S as base any set of blocks. 

 Let us denote the length of the block at the point x by X, and, for simplicity, let us 

 first consider X as being always positive. Then, if we divide S in any manner into a 

 finite or countably infinite set of measurable components, S 1( S 2 ..., and at each point 

 of S, we replace the given block by one of length equal to the upper limit of the 

 lengths of the blocks at points of S,, we get a new set of blocks which, regarded as a 

 plane set of points, contains the given set of blocks. Since the sum of a finite or 

 countably infinite series of non-overlapping measurable sets is a measurable set whose 

 content is the sum of the contents of its components, it follows that the new set of 

 blocks constitutes a plane set of points which is measurable, and has for content, by 

 the Lemma, the upper summation of X over S corresponding to the mode of division 

 adopted. Hence the outer measure of the content of the plane set of points 

 constituted by the given set of blocks is not greater than this upper summation, and 

 therefore, since this is true for every mode of division, not greater than the outer 

 measure of the integral of the blocks over S. Similarly for the lower measure of the 

 integral. 



If, on the other hand, X be sometimes positive and sometimes negative, we must, 

 as LEBESGUE does, consider separately the positive and negative blocks ; thus we get 

 the following theorems : 



Theorem 25. An upper or loiver summation is the difference of the contents of two 

 measurable sets of blocks, one on the positive and one on the negative side of the axis 

 of x. 



Theorem 26. Given any set of blocks on a measurable set S, the outer measure of 

 the content of the plane set of points constituted by tfie positive blocks minus tlie inner 



