248 DR. W. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 



Set S. If the whole set of blocks in measurable, the Junction is said to be intcgrable 

 in the generalised sense (or summable) with respect to the fundamental set S. In this 

 case the positive blocks alone form a measurable set, and so do the negative blocks, 

 and the excess of the content of the positive over that of the negative blocks is called 

 the generalised integral of the function with respect to S. 



LEBESGVE (p. 255) has occasion to use integration with respect to a measurable set, 

 but only in the case when the set is contained in a finite segment. The mode of 

 definition adopted by him is that of completing the function by ascribing to it the 

 value zero at all points of the segment other than those of the set in question, and 

 then defining the integral of the function with respect to the fundamental set as 

 equal to that of the extended function in the whole segment. In this case the 

 generalised integral so defined is evidently the same as that defined above and in 

 24 ; the mode of definition is, however, open to some objection ; characteristic 

 properties of the function such as continuity or semi-continuity with respect to the 

 fundamental set, which materially simplify the properties of the integral, are not 

 maintained by the extended function ; on the other hand the definition suggests a 

 difficulty, when dealing with fundamental sets of finite content not lying in a finite 

 segment, which is entirely illusory, and gives a pre-eminence to the finite segment as 

 region of operation which it does not in reality in any way possess. 



29. LEBESGUE'S theorem that the sum of two summable functions is a summable 

 function, and its integral is the sum of their integrals, is equally true when the 

 fundamental set is any measurable set of finite content. It is an immediate result of 

 any of the definitions that this is the case when one of the functions is a constant, 

 thus, the upper and lower limits being as usual finite, if the theorem is true for 

 positive functions it is true always. In the case of positive functions the theorem is 

 geometrically equivalent to the following ; 



Theorem 27. Given any set of positive blocks forming a measurable plane set of 

 points, the blocks may be shifted up parallel to themselves without altering the content 

 of the set, provided the amount of shifting at each point x is a summable function of x. 



Divide the base S up into any number of measurable 

 components Si, S 2 , ... and consider the set in the shifted 

 position. At any point x of S ; let the block be PQ, so 

 that Pa? represents the amount of shifting from the position 

 when the lower extremities were all on the a;-axes. Prolong 

 the block to R, so that PR is equal to the upper limit of 

 the length of the blocks corresponding to all points of S,. In 

 x this way we get a plane set containing the given set. 



Now, by Lemma 1, 27, the content of a set of blocks each 



of length PR, erected at all the points of S,, is PR . S,. Hence, since we may 

 evidently consider Rx as having been obtained by shifting up the ordinate Px the con- 

 stant amount PR, the content of the set of blocks Rx = that of the blocks Px + PR.S,. 



