CoNRAN- — The Kiemanii Integral and Pleasurable Sets. 7 



A closed domain is a domain together with its boundary points. 



A finite or enumerably infinite number of open domains form an open 

 domain. 



An open domain is measurable in Lebesgue's* sense, and its content or 

 measure is identical with the number which Jordanf calls " I'etendue 

 interieure." An open domain of the first or second kind may be divided 

 (e.g., by parallels to the axes) into a finite number of sub-domains of 

 assignably small diameters, such that the common boundary-points of adjacent 

 parts have zero plane content. If each sub-domain be divided in like manner, 

 and this process be continued indefinitely so that the greatest diameter in the 

 7'th stage diminishes indefinitely with r, I shall call this a normal sy.stem of 

 subdivisions. Tlie content of an open domain so divided is equal to the sum 

 of the contents of the normal sub-domains at any stage. 



Double Integral in an Open Domain. 



8. Let /(.-);, y) be a function of two variables defined at all the points of 

 an open domain D. I define the upper integral in D as follows : — • 



Let D be divided by any normal system of sub-division, and let 

 D,.,, D,i, . . . Dr,i be the sub-domains of the 7'th stage, and CVi, On, . . . C,„ 

 their plane contents. Let the upper limit oif{x, y) for all the points in D,-^ 

 be Urs- Then the limit of the sum 



S=7t 



as r increases indefinitely is the upper integral. 



That this limit exists and is unique follows by precisely similar reasoning 

 to that employed in establishing the limit as usually defined. 



The lower integral is defined in an analogous manner as the 

 limit V 7- r' 



where Z,-s is the lower limit of the function for all internal points of Drs- 

 For domains of the first kind there is no difficulty in proving, by an 



adaptation of the method in § 1, that the above definitions are equivalent to 



Jordan's.J 



Cor. 1. — If an open domain of the first kind Di be a component of any open 



domain D, the points of D which are external to i>, form an open domain A, 



and the 



content of D = content of Di + content of D^, 

 and . 



(Zer) '''^'^^^ ^" ^ = (lower) '''''^'^^ ^" ^^ + (ZerJ "^^^^^'^^ ^^ ^^ 

 * Lecons sm rintegKition. GuutliierVillars. 1904. t Coursd' .Analyse, vol. i. 1909. | Loe. cit, 



