Is there a good place to begin studying compact embeddings into the space of measurable functions?

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Rogier Brussee

Measurable functions are awkward to work with, because you cannot change things on a set of measure zero and there is no natural topology on them (that I know of). In any case you might want to study the Sobolev embedding theorems and variants thereof first.

Mieczyslaw Cichon

In my opinion, you can start with the book by Martin Vaeth "Ideal Spaces" LNM 1664. But, of course, you should think about classes of measurable functions...