2 Proceedings of the Royal Irish Academ/j. 



I have extended the theorem coneeming the evahiation of a multiple 

 integral hy repeated simple integrations to the ease when the region of 

 integration has a frontier of positive content. 



The Borel-Lebesgue theory of content is adopted throughont. 



The function and the region of integration are in all cases taken to lie 

 finitely bounded. 



The Integral in a Continuous Interval. 



1. Suppose the upper integral of f{x) in the open interval a <.r<b to 

 be defined in the following manner : — 



Divide the interval into n parts at the points a - x^,, .r,, x„ = h. 



Let My be the upper limit of f{x) in the open interval x,-<x<Xi-^i, and 

 form the sum 



' S'i>/,.(.'r„, - ,^v). 



If each elementary interval be di\dded in like manner, and the corre- 

 sponding summations formed, and if this process be continued indefinitely, 

 the successive summations form a monotone sequence. Let J be the limit 

 of this sequence ; then clearly J <:I, where / denotes the ordinary upper 

 integral 



It is not difficult to prove that ./ = /. 



Let the points XiX. ... &c., be so chosen that 



2if,.(AV„-.r,.)- J"<£, 



where e is any assigned positive quantity. 



Now choose a positive quantity S so that 2S is less than the least length 

 of the intervals («,., ccr^\), and let M',- denote the upper limit of f{x) in the 

 closed interval x,. + S <: .r $ .r,.^, - S. Then M,- > M',-, therefore 



2i)i;.0r,.„ - X,)>^M'r{Xr,, - Xr - 2g), 



and -2^1' r{x,.,, - Xr - 28) + 2n.M.^> I, 



where M is the maximum value of i/(.r)|. 



.-. J+ t>I-2nM.^; 



.". J>I, and we have also I>J; 



.: J=I. 



The definition of the lower integral may be similarly modified without 

 altering its value. 



