Fine properties of functions from Hajasz-Sobolev classes Mαp, p > 0, I. Lebesgue points

Let X be a metric measure space satisfying the doubling condition of order gamma > 0. For a function f a L (loc) (p) (X), p > 0 and a ball B aS, X by I (B) ((p)) f we denote the best approximation by constants in the space L (p) (B). In this paper, for functions f from Hajasz-Sobolev classes M (alpha) (p) (X), p > 0, alpha > 0, we investigate the size of the set E of points for which the limit lim (r ->+0) I (B(x,r)) ((p)) f = f*(x). exists. We prove that the complement of the set E has zero outer measure for some general class of outer measures (in particular, it has zero capacity). A sharp estimate of the Hausdorff dimension of this complement is given. Besides, it is shown that for x a E Similar results are also proved for the sets where the "means" I (B(x,r)) ((p)) f converge with a specified rate.