To characterize compact parts of spaces $L^p(\Omega)$, we introduce a concept of equi-integrability based on the approximation of elements of $L^p(\Omega)$ by simple functions. The resulting theorem will be used to develop a new methodology to prove and extend results about the compactness of Sobolev embeddings.
"A Characterization of Compact Parts of Lp Spaces Application to Sobolev Embeddings." Real Anal. Exchange 27 (2) 683 - 710, 2001/2002.