I am looking for a list of classical spaces which are known to compactly embed into the space of measurable functions and any which are known NOT to compactly embed into L_0. L_0 is just the space of measurable functions over the interval (0,1) equipped with Lebesgue measure.
I believe you consider L^0 equipped with the metric of convergence in measure. Then the first answer to your question is that NO L^p space is compactly embedded into L^0. Indeed, consider a sequence f_n, constructed as follows:
the segment (0,1) is split into 2^n intervals of equal length, f_n=1 on odd intervals and f_n=-1 on even intervals (flipping sign function). Obviously, |f_n| =1 at every point so
|| f_n ||_{L^p} = 1 for all n and p.
On the other hand, \int_0^1 f_n ds =0 and it is easy to verify that
\int_0^1 f_n s^m ds < c 2^{-n} for all m =1,2,3,...
Since polynomials are dense in continuous functions, we conclude that
the measures f_n ds converge to zero. So an L^0-convergent sub-sequence f_k of f_n converges to 0 if exists.
However, | { |f_n| =1} | = 1, so no sub-sequence of f_n converges to 0 in L^0.
I believe, neatly approximating f_n by continuous functions one can show that C(0,1) does not compactly embedded into L^0 either. Probably, even the harmonics f_n(s) = sin( \pi n (s-1/2 ) ) would do.
On the other hand, L^p(0,1) and C(0,1) are continuously embedded into L^0(0,1). Hence spaces compactly embedded in L^p(0,1) and C(0,1) - such as Sobolev and Hoelder spaces and functions of bounded variation - are compactly embedded into L^0.
It seems all about classic functional spaces.
Concerning natural embeddings of Lp[0,1] into L0[0,1] there is a striking result that on some infinite-dimensional subspaces the Lp topology is the same as the convergence in measure topology. An example of such a subspace is the linear span of Rademacher functions; another example is the subspace spanned by the lacunary trigonometric sequence $$\sin(2^n\pi t)$$, n = 1,2, ...
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