CoNRAN — The Eirmanu tnlegrnl ami Measurable Sets. 13 



The only case of interest in the Eiemaun integral is when the function is 

 integrable in the given set. In this case the integrals defined in this paper 

 are equal to the Lebesgue integrals. 



Subject to the condition A stated above, the determination of the integral 

 in a measurable set of any number of dimensions has been shown to depend 

 ultimately on simple integrations in linear intervals. 



Exaviplel. To evaluate xdx over a measurable set jS' of linear content ?/, 



and symmetrically disposed in the interval < x < a. 



Case 1. Suppose S is closed, and that the complementary set of open 

 intervals are oi, a.i, an, a_2, . . . &c., where a,- and a_,- are equal in length 



and equidistant from the point x = - • 



Let the end-points of or be ,(',. and x'r. Then 



xdx = -i {x't? - x/), 

 J «>■ 



and X dx = i (/_,.- - a;_r'), 



X dx = i (x'r - Xr) {Xy + x' ,■ + X.r + J-'-r) 

 J O.r + ttr 



= |- a X content of (or + a.,-), 

 xdx = iV « X content of the set of intervals fal, 



J {a} 



X dx = X dx - \a (a - J") = | a J. 



Case 2. Let S be an ordinary inner limiting set, and let the defining sets 

 of intervals be {a}i, {aj^, . . . &c. Then 



f xdx= ^^1 f xdx= li°^ r^contentof |ai„l =.V«/. 



Case 3. If (S' be any measurable set, we can determine an inner limiting 

 set S', containing S and having the same content as S. 



= h O.J as 

 is ]s' - 



before. 



Examiie 2. The integral of a discontinuous function over the set of its 

 points of continuity. 



Fio. 2. 



