6 Proecnh'iijifi nj Ihe Roi/al Irish Acitileiiu/. 



Now let. ,/' (.!•) be deKned at all the points in a fundamental suguiont 

 AB of length L, and let /C denote the set of points in AB at which the 

 saltiis 5" e,„ where e„ is one of the numbers of a monotone positive sequence 

 Ci, C2, . . . &c., having the limit zero. Then Kn is a closed set, and the sequence 

 Ki, K^, . . . &c., defines an ordinary outer limiting set consisting of the points 

 of discontinuity. Now let U„ be the set complementary to Kn with respect 

 to the fundamental segnient AB. E„ consists of a set of non-overlapping open 

 intervals (a trivial exception occurs when A or 5 is a point of continuity), 

 and the sequence E^, E^, . . . &c., defines an ordinary inner limiting set con- 

 sisting of the points at which the function is continuous. Let this be denoted 

 by E, then 



and = lim 



J-£',. JE n^^v. 



En Ji'n 



I = lim [ 



Moreover, by the lemma 



<; In y~ content of E,, < e,j Z. 



Hence = which proves the theorem. 



JE [e 



Cor. — From | 5 it is clear that the addition or removal of a set of zero 

 content will not affect the values of the upper and lower integrals in a given 

 measurable set. Hence a sufficient condition for integrability is that those 

 points of the set at which the function is discontinuous should form a set of 

 zero content. That this is also a necessary condition is immediately evident. 



The BouMc Integral. 



7. The double integral is usually defined for a closed region whose 

 boundary points have positive plane content. Por my purpose it is necessary 

 to define the integral for an open region which may have a boundary of 

 positive content. 



The region of integration will be, in the first instance, what Baii'e* calls 

 an open domain — that is, a set of points such that each point is the centre of 

 a circle which contains in its interior and on its boundary none but points of 

 the given set. 



Prom the point of view of integration it is important to distinguish two 

 types of domains — those whose boundary points have zero plane content, 

 which I will call domains of the first kind, and those whose boundary points 

 have positive plane content, which I will call domains of the second kind. 



