Not a formal proof, but a heuristics why it is very unlikely that such a characterization exists: Measurability is an assumption about preimages, while the separability assumption you require is on the image.

@GeraldEdgar: The name you mention is spelled W.A.J. Luxemburg

I do not think that you need a strictly increasing sequence with strict inclusions. And in fact that "strict" need not hold in metrizable spaces (e.g. if $Y$ is finite).

Thank you. So the argument is essentially the same than proving/using Hartog's theorem (for injections) or the analogous Lindenbaum theorem (for surjections). These are not hard, but I was hoping for an essentially simpler proof.

for any set $S$ the class of ordinals $\alpha$ such that $\alpha$ injects into $S$ (resp. $S$ surjects onto $\alpha$) is a set. Is there a simple argument to see this (especially for the "surjects" part)?

It is one variant of a straightforward condition ensuring that the operator $A$ on the right-hand side maps $S=\{x\in C(I,E):\lVert x(t)\rVert\le r(t)\}$ into itself, and so Schauder/Darbo/whatever fixed point theorems are applied in many papers for $A\colon S\to S$. A trivial artificial example is for a measurable kernel satisfying $\lVert k\rVert_\infty\le C$ a function $f$ satisfying $\lVert f(t,u)\rVert\le M$ for $\lVert u\rVert\le CM$: Put $r(t)=CM$, $a(s)=1$, and $b(u)=M$ for $\lVert u\rVert\le CM$. Thus, formally no growth hypotheses needs to be satisfied for $f$ w.r.t. $u$.

BTW, the equivalence of the problem with the Istratescu measure of noncompactness is not as trivial as it might seem. I did not recall that when I wrote the above comment, but now I recalled that I had once given a rigorous proof of this equivalence myself (formula (3.6) in my monograph on topological analysis). There are quite some publications on the Istratescu mnc which are not translated into English and not electronically accessible; it is probably necessary to contact one of the experts (e.g. Nina A. Erzakova).

I don't know, but there is quite an active community around the Istratescu measure of noncompactness. It might be a well-known result for them.

Isn't the question equivalent to ask about the Istratescu measure of noncompactness of the unit ball $M$ of $L^q$ in the space $L^p$? (Literature in that formulation might be easier to search for.) Recall that the Istratescu measure of noncompactness of $M$ is defined as the supremum of $\inf_{n\ne m}d(x_n,x_m)$ over all sequences $x_n\in M$.