  Let $X$ be a quasi-Banach function space over a doubling metric measure space
$\mathcal P$. Denote by $\alpha_X$ the generalized upper Boyd index of $X$. We
show that if $\alpha_X<\infty$ and $X$ has absolutely continuous quasinorm,
then quasievery point is a generalized Lebesgue point of a quasicontinuous
Haj{\l} asz function $u\in\dot M^{s,X}$. 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--Haj\l asz, Orlicz--Haj\l asz and variable
exponent Haj\l asz functions.
