Let σ be a finite positive singular Borel measure defined on Euclidean space RN. For w ∈ RN and y > 0, its Poisson integral is defined by the formula
where CN is chosen so that
Since σ is singular, almost everywhere with respect to Lebesgue measure on RN. On the other hand, almost everywhere dσ. It follows that for all sufficiently small y,
is a non-empty open subset of RN. If σ has compact support then |Ey| → 0 as y → 0, where |Ey| denotes the Lebesgue measure of Ey. In this paper we give a lower bound on the rate at which |Ey| may go to zero. The lower bound depends on the smoothness of the measure; the smoother the measure, the more slowly |Ey| may approach 0.