CoNRAN — The Hiemann Integral and Measurable tiets, 9 



repeated integrations has been discussed by Jordan. In this case also the 

 method of extending the region of integration so as to include the whole 

 fundamental rectangle, the value zero being assigned to the function at points 

 not belonging to the set, is valid. 



Frontier of positive content.— In this ease it becomes necessary to evaluate 

 the double integral in a domain of the second kind. Jordan's method does 

 not apply nor is it justifiable to extend the region so as to include the whole 

 fundamental rectangle. 



The question then arises — " Under what circumstances may the double 

 integral in an open domain of the second kind be found by repeated single 

 integrations ?" "We must first express the plane content of such a domain as 

 a repeated integral. 



12. Lemma I. — The linear content of an ordinal section of a plane open 

 domain is a lower semi-continuous function of the abscissa. 



Let the domain be contained in a funda- 

 mental rectangle ABCD. 



Let / («) be the content of the ordinal 

 section at the point x. 



This section consists of a set of non-over- 

 lapping open intervals. Select a finite number 



of these of total length > I {x) - -, and 



shorten each of these intervals so that the 

 curtailed intervals have total length > /(.r) - 1. 



Each point of the curtailed intervals is an internal point of the domain, and 

 is therefore internal to a rectangle consisting only of points of the domain. 

 By the Heine-Borel theorem, a finite number of these rectangles suffice to 

 cover all the points of the curtailed intervals. Therefore, with x as centre, 

 an interval fo - S < x' <x + ^ can be constructed such that the ordinate 

 section at x' has content > I{x) - i. Hence / («) is lower semi-continuous. 



Lemma 2. — If / {x) be > k for all points in the interval a <x < j5, the 

 content of the part of the domain bounded by the ordinates at a and j3 is 

 > K (/3 - a). Consider the points on the a^-axis in the closed interval 

 a+£^x^j3-e. From Lemma 1 it is evident that each point x is the 

 centre of an interval of length 29 such that rectangles of area > 23 . k are 

 contained in the corresponding part of the domain. By the Heine-Borel 

 theorem a finite number of these intervals cover all the points a + e<x<l3-£. 

 Therefore there is a total rectangular area > k (/3 - a - 2£) contained in the 

 given domain, where t may be as small as we please. 



This proves the lemma. 



IS.I.A. VROC, VOL. XXX., SECT. A. [2] 



