1904.] 



The General Theory of Integration. 



449 



Lebesgue's theorem that the sum of two summable functions is a 

 summable function and its integral is the sum of their integrals is 

 then proved by geometrical considerations, and a more general theorem 

 is given, viz. : — 



If X° and X 1 be the outer and inner measures of the content of the 

 ordinate section of a measurable set by the ordinate through 

 the point x, X° and X 1 are both summable functions, and the 

 generalised integral of either is the content of the measurable set. 



It is here assumed that the content of the set got by closing the 

 measurable set is finite. The content of any measurable set, with 

 this restriction only, is thus obtained in the form of a generalised 

 integral and, therefore, of an ordinary integral ; in fact — 



The content of any measurable set (provided the set got by closing 

 it has finite content) is j Idx. 



Here I is the content of the component of the fundamental set at 

 which the inner (or the outer) measure of the content of the ordinate 

 section of the given set is greater or equal to Jc. 



It is to be remarked that though in this abstract reference has only 

 been made to linear and plane sets and to the corresponding integrals, 

 the arguments are perfectly general and apply to space of any number 

 of dimensions. For instance the concluding result is as follows : — 



To find the content of a measurable %-dimensional set, take any 

 hyperplane section and project the whole set on to this hyper- 

 plane. Any measurable set containing this projection we take 

 as the fundamental set S. Divide S up in any way into a finite 

 or countably infinite set of measurable components, and multiply 

 the content of each component by the upper (lower) limit of 

 the values of the (linear) inner or outer content of the corre- 

 sponding ordinate sections of the given set, summing all such 

 products, the lower (upper) limit of all such summations is the 

 content of the given set. 



