DR. W. H. YOUNG ON THK :KNKRAL THEORY OF INTEGRATION. 247 



measure of the content of the negntin- block.* in not greater than the miter measure of 

 the integral of the block* over S. 



Similarly, the inner measure of the content of the positive block* minus the outer 

 measure of that of the iiegatiiv block* is not less than the inner measure of the integral. 



In symbols 



Ef-Ei'sP, Ei'-E, *!', 



where E, and E, are respectively the set of points constituted by the positive and the 

 negative blocks, and I denotes the integral, the indices o and i denote outer and inner 

 measures respectively. 



Denoting the sum of EI and E a by E, and supposing the upper and lower limits of 

 the lengths of the blocks to lie finite, EI and ES are the common parts of E, and the sets 

 of blocks on S as base whose heights are respectively the upper and the lower limits of 

 the lengths of the blocks, both these sets of blocks are measurable. Thus we see that, 

 if E is measurable, E t and E are so also, so that I" and I 1 are equal to one another 

 and to the difference of the contents of EI and ES, that is to say, a summable 

 function is integrable in the generalised sense of 24, and the integral as defined in 

 LKBKSOUE'S geometrical manner is the integral as defined in 24, S being a finite 

 segment. 



Conversely, if I" and 1' are equal, 



E^-E,' S I S Ei'-E/, 



but since the outer content of a set is not less than the inner content, 



E,'- E,' i EI' - Ej, whence E,- Ej 1 = ,''-,. 

 Therefore 



E,"+E 4 = E.' + E, 1 , 



and therefore, since the outer measure of the content of E is not greater than Ei'+E./, 

 and the inner measure is not less than E,' + Eg', the inner and outer measures 

 of the content of E are equal, and E is a measurable set. Thus a function which is 

 integrable. icith respect to a finite segment in the generalised sense of 24 is a 

 summable function, and its integral is the Lebesgue integral, by LEBESGUE'S geometrical 

 definition. 



Summing up the results of this section, we have the following theorem : 



Theorem. Lebesgue integration is identical with generalised integration tvith 

 respect to a finite segment. 



28. In general, without confining our attention to finite segments, we have from 

 what has l>een shown the following geometrical definition of generalised integration 

 with respect to a measurable set 8 : 



Geometrical Definition. At. each JKH'/J x of a measurable set S draw an ordinate 

 (block) equal to the value of a- function defined for every point of S, the outer measure 

 of the content of the positive blocks minus the inner measure of the negative blocks is 

 called the outer measure of the integral function with respect to the fundamental set. 



