A criterion for a set to be of 1-dimensional measure zero
Tadashi Kuroda · Japanese journal of mathematics · 1959
Let F be a domain in the complex z-plane ƒ ¶ and let E be the boundary of F. We assume that the point z=• ‡ is an interior point of F. Consider a finite number of doubly connected domains R(k)(k=1, •c, v) satisfying the followings: i) the closure R(k) of R(k) is contained in F, ii) the boundary of R(k) consists of two analytic closed curves, iii) the complementary set of each R(k) consists of two domains the one F(k) of which contains the point z=• ‡ and the other G(k) of which has at least a point common with the set E, iv) any point of the set E is contained in some G(k) and v) if i•‚j, then R(i) lies in F(j).We shall say that the system R={R(k)}vk=1 of such doubly connected domains separates, in F, the point z=• ‡ from E.If we put F0=•¿vk=1F(k),we can see that F0 is a domain compact relative to F and that the complementary set of the set