Let LSC = LSC([0,1]) be the set of non-negative, lower semi-continuous functions on the unit interval which take values in $\mathbb{R}_+ \cup \{\infty\}$. We use epigraph topology on LSC, i.e. a sequence of functions converges iff the sequence of epigraphs converges in Fell (or equivalently Kuratowski) topology. This convergence coincides with $\Gamma$-convergence (i.e. $f_n\to f$ iff for all $x_n \to x$ we have $\liminf f_n(x_n)\ge f(x)$ and for all $x$ there exists $x_n\to x$ with $\lim f_n(x_n) = f(x)$). LSC with this topology is a compact, metrisable space.

Now consider the subspace of bounded functions in LSC. This is clearly a countable union of closed subsets, hence Borel-measurable in LSC. Now my question is:

Is the subset of $\mathbb{R}_+$-valued functions in LSC Borel-measurable?

Edit: I am asking this, because I naturally came to the space of semi-continuous functions that are *not* allowed to take on the value $\infty$, and need a "nice" topology on it. Epigraph topology makes the things converge which I want to converge, but being separable, metrisable is just not "nice" enough, while being a measurable subset of a compact metric space (= Lusin space) would do. Maybe continuous image of a Polish space (= Souslin space) would also be enough.

So if anyone had, alternatively, an idea of a "similar" topology with better properties, this would also be great!

P.S. I found out that on the space of $\mathbb{R}_+$-valued functions in LSC, epigraph topology is also equivalent to convergence of the epigraphs in Hausdorff metric (which is not the case on the whole of LSC). Unfortunately, Hausdorff distance of the epigraphs is also an incomplete metric.