Let $X$ be a locally convex topological vector space, and let $f:X\to\mathbb R\cup\{\infty\}$ be a proper, convex, lower semicontinuous function, whose effective domain $D:=f^{-1}(\mathbb R)$ is compact. Let $S\subseteq D$ be the set of points at which $f$ is subdifferentiable.

My question is somewhat vague: What results are there that say $S$ constitutes "most of" $D$? One example of such a result is the Brøndsted-Rockafellar theorem, which says $S$ is dense in $D$ whenever $X$ is a Banach space.

Ideally, I'd like to know any results guaranteeing $S$ is most of $D$ in some sense other than topological (e.g. measure-theoretic). Any pointers to such results would be very helpful.

shy; a set that isn't shy is calledprevalent. $\endgroup$prevalentif its complement is shy. $\endgroup$