Some Remarks on the Measurability of Certain Sets

The present note contains some elementary remarks on sets defined by simple geometric properties. Our main tool will be the Lebesgue density theorem. First we introduce a few notations : d(a, b) denotes the distance from a to b and Six, r) the open sphere of center x and radius r. A point x of a set A is said to be of metric density 1 if to every e there exists a ô such that AC\S{x, r) , r < 5, has measure greater than (1 — e) times the volume of S(x, r). 'A denotes the closure of A. (1) Let E be any closed set in w-dimensional euclidean space. Denote by Er the set of points whose distance from E is r ( r>0 ) . We shall prove that Er has measure 0. The set Er is clearly closed and therefore measurable. If it had positive measure it would contain a point of metric density 1. Let x be any point of Er and yÇzE be one of the points in E a t distance r from x. Then S(y, r) cannot contain any point of Er. Thus x cannot be a point of metric density 1, which completes the proof. This proof is due to T. Radó. (2) Let A be any set of measure 0 on the positive real axis. Denote by E A the set of points whose distance from E is in A. We shall show that EA has measure 0. As is well known A is contained in a G$, say G of measure 0. Thus it suffices to show that E0 has measure 0. Eg is clearly a Gs and thus measurable, so that again it will suffice to show that Eg has no point of metric density 1. Let x be any point of Eg and y any one of the points of E closest to it. Denote by Cx(yi, 772) the half cone defined as follows: z G C ^ i , 772) if d(z, x)0 . Let ;yi 0 such that the upper limit of the difference quotient of d(z, yi) with respect to d(z, x) is less