The essential boundary of certain sets

The essential boundary of a measurable set is related to the de Giorgi perimeter and was introduced by Vol'pert in his "improvement" of Federer's work. For a totally disconnected compact set of positive measure in n space the essential boundary can be of Hausdorff n 1 dimension but cannot have a finite (n 1)measure. Let E c R'" be a measurable set with respect to Lebesgue measure. It is said to be of finite perimeter if all the partial derivatives t1(A),... I, Pn(A) of its characteristic function X are totally finite measures over the Borel sets A C R . It is well known that ti = pi(Rn)= JvI, where vi is the infinum of the variations in xi of all functions equivalent to X and the integration is over the (n 1)-space orthogonal to the xi axis, Oxi. The value of the perimeter is then the variation measure of the vector valued measure (t1(A), ... ., n(A)), evaluated for Rn. It was shown by Federer [1] that the perimeter is equal to the (n 1)-measure of a set that he called the reduced boundary of E, consisting of those points at which a certain generalized normal exists. Specifically, a point p is in the reduced boundary of E if there is an (n 1)-plane v through p such that the part of E on one side of v has density 0 at p, and the part of CE on the other side of v has density 0 at p. (The k-measure X k(E) of a set in Rn will mean the k-dimensional Hausdorff measure, normalized so that the k-dimensional unit cube has measure 1.) Vol'pert [2] showed that in the result the reduced boundary may be replaced by the essential boundary ae(E), consisting of those points of Rn which are neither points of density 1 nor points of density 0 of E. Clearly, the essential boundary of E contains the restricted boundary. Vol'pert's remarkable theorem asserts that, if E has finite perimeter, then the Hausdorff (n 1)-measure of the restricted and essential boundaries are equal. The Lebesgue theorem guarantees that ae(E) is of n-measure zero, but in its perimeter role ae(E) has more of an (n 1)-dimensional flavor, and evidently coincides with the ordinary boundary a(E) when this is a sufficiently smooth surface. Going to the opposite extreme, in this note we shall discuss the possible nature of Me(F), with respect to (n 1)-measure, when E is a nowhere dense set of positive measure in R'". Received by the editors October 23, 1984. 1980 Mathemnatics Subject Classificcation. Primary 28A75. 1 These authors were supported in part by grants from the National Science Foundation. Part of the work was accomplished while the third author was visiting the University of California at Santa Barbara during a Special Year in Real Analysis. :'?1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page