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



we have seen, functions which are integrable in the old sense are integrable in the 

 new sense, and the integrals agree, but the converse is not true. In the case of an 

 upper (lower) semi-continuous function, the generalised integral is the upper (lower) 

 integral. 



25. LEBESGUE gives two definitions of his generalised integral, which I shall, for 

 convenience, allude to as the Lebesgue integral. 



The first is a geometrical definition, and has the disadvantage that the positive 

 and negative values of the function have to be considered separately. It is as 

 follows : 



Geometrical Definition of the Lebesgue Integral. Let f be a function defined for 

 every point of a finite segment (a, b) ; consider the plane set of points defined by the 

 three inequalities* 



Let E! and E s be the two parts of E respectively above and below the axis of x 

 (the points on the axis of x may be considered as belonging to whichever we prefer 

 EX or E 2 ). If E is a measurable set, then both EI and E 2 are measurable, t In this 

 case the function f is said to be summable, and the excess of the content of E! over 

 that x>f E 2 is defined to be the Lebesgue integral of f over the segment (a, b). 



LEBESGUE'S second definition is analytical. 



Analytical Definition of a Summable Function. A summable function is such that 

 the set of values of x, for which the values of the function lie between any two 

 quantities a and b, is measurable. Conversely, if this condition is satisfied, and the 

 upper and lower limits of the function are finite, the function is summable. 



Analytical Definition of the Lebetgue Integral, Let the region of variation of f(x) 

 be denoted by (k 0> k A ), and let it be divided into n parts each less than a, say, at the 

 point h, k. 2 , ...&_,. 



Let f denote the content of those points x at which f=kj; and e'i that of the 

 points x at which k i -\<.f(x)<.k i . 



Then it may be shown that the two summations 



tkf i + 'k i _ l <f i and 2^, + S^', 



01 01 



have a common limit, when a is decreased indefinitely ; this limit is the Lebesgue 

 integral of f(x) from a to b. 



The identity of the two definitions is easily proved ; in the case of a function which 

 is always positive, the former of the two given expressions evidently represents the 



* Here I have corrected an obvious misprint in LEBESGUE'S paper. 



t This is stated without proof by LEBEKGUE in 17 ; it is a special case of the theorem of 18, p. 251. 

 A more simple proof is afforded by considering EI as the common part of E and a sufficiently large 

 rectangle on the given segment as base and lying on the positive side of the z-axis, and E 2 as the difference 

 between E and EI ; since the common pirt or the difference of two measurable sets is measurable. A 

 similar proof applies to the theorem of 18, given below here as the analytical definition. 



