Hausdorff Measures and Lebesgue Points for the Sobolev Classes $W^p_\alpha$, $\alpha>0$, on Spaces of Homogeneous Type

Suppose that $(X,\mu,d)$ is a space of homogeneous type, where $d$ is the metric and $\mu$ is the measure related by the doubling condition with exponent $\gamma>0$, $W^p_\alpha(X)$, $p>1$, are the generalized Sobolev classes, $\alpha>0$, and $\operatorname{dim_H}$ is the Hausdorff dimension. We prove that, for any function $u\in W^p_\alpha(X)$, $p>1$, $0<\alpha<\gamma/p$, there exists a set $E\subset X$ such that $\operatorname{dim_H}(E)\le\gamma-\alpha p$ and, for any $x\in X\setminus E$, the limit
$$ \lim_{r\to+0}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}u\,d\mu=u^{*}(x) $$
exists; moreover,
$$ \lim_{r\to+0}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}|u-u^{*}(x)|^{q}\,d\mu=0,\qquad \frac{1}{q}=\frac{1}{p}-\frac{\alpha}{\gamma}. $$
For $\alpha=1$, a similar result was obtained earlier by Hajłasz and Kinnunen in 1998. The case $0<\alpha\le1$ was studied by the author in 2007; in the proof, the structures of the corresponding capacities were significantly used.