Generalized Lebesgue Points for Hajlasz Functions

Let X be a quasi-Banach function space over a doubling metric measure space P. Denote by alpha(X) the generalized upper Boyd index of X We show that if alpha(X) < infinity and X has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajlasz function u is an element of(M) over dot(s,X) Moreover, if alpha(X) < (Q+s)/Q, then quasievery point is a Lebesgue point of u. As an application we obtain Lebesgue type theorems for Lorentz-Hajlasz, Orlicz-Hajlasz, and variable exponent Hajlasz functions.