The Luzin approximation of functions from the classes $W^p_\alpha$ on metric spaces with measure

In this paper we prove an analog of the Luzin theorem on correction for the Sobolev-type spaces on an arbitrary metric space, whose measure satisfies the doubling condition. The correcting function belongs to the Hölder class and approximates a given function in the metrics of the initial space. Dimensions of exceptional sets are evaluated in terms of Hausdorff volumes and capacities.