Sobolev spaces on an arbitrary metric measure space: Compactness of embeddings

We formulate a new definition of Sobolev function spaces on a domain of a metric space in which the doubling condition need not hold. The definition is equivalent to the classical definition in the case that the domain lies in a Euclidean space with the standard Lebesgue measure. The boundedness and compactness are examined of the embeddings of these Sobolev classes into $L_q$ and $C_\alpha$. We state and prove a compactness criterion for the family of functions $L_p(U)$, where $U$ is a subset of a metric space possibly not satisfying the doubling condition.