528 KELLOGG. 



f- -J. S„ is obtained from S„_] by the removal of the open middle 



thirds of the 2" l equal intervals of which S n -i consists. The set S 

 is the points common to all the sets S n as n runs through all positive 



integers. 



Let K be a circle containing S in its interior, say the circle 

 ,r + if = 4. The region, T*, in question, consists of the points in- 

 terior to K, excluding the points of S. The boundary points of T* 

 within K then form a perfect set of Borel measure 0. Hut in spite 

 of the relative sparseness of the points of S and the infinite connectiv- 

 ity of '/'*, this region belongs to the class [7\]. 



3. The Lebesgue Barrier Sits. 



This assertion may he proved by means of the notion of barrier 

 functions, introduced by Lebesgue. 2 Suppose that there exists, for 

 each boundary point, />'. of a given region. '/', and for every positive 

 number, t. a function, Us, t I' • with the following properties: it is 

 harmonic in 7 , and assumes at every point, /', of 7 , a value not less 

 than the distance />'/'. As /' approaches />, the function approaches a 

 number less than t, or more generally, there exists a neighborhood, N, 

 of B, such that when /' is a point of T in N, ^ Us, t (P) = *• A set 

 of such functions, we call, after Lebesgue, a harrier set for the region 7'. 

 Its existence is a necessary and sufficient condition that T belongs to [Ti]. 

 While this theorem, as well as its demonstration, is implied in the 

 report of Lebesgue, the reader may welcome the following more 

 explicit proof. 



First, -Mipptw that the harrier set exists for T. Wv approach the 



region '/' by a set of regions, /,, /. : , / ;{ each containing the preceding 



and hounded by a Unite number o'i regular curves, and such that each 

 point of T is interior to one of them, and hence to all later ones. Let 

 us supposr first that the boundary values /•' are those of a polynomial, 

 F(x,y). Then harmonic functions exist for the regions t u t- : , tz,. .., 

 which approach the same values as /•' on their boundaries. These 

 functions will at least contain a sequence, [U n ], which converges uni- 

 formly in any closed subregion of '/' to a harmonic function ('. That C 

 will approach the required boundary value at a boundary point, B, 

 may he seen as follows. Let L he an upper hound for the difference 



3ur k> probleme de Dirichlet, C. R. vol. 154 (1912). p. Go. - .. 



