Theorem 21.11.177E0B71CF3B0E

Proof. Apply the closure theorem for proper convex functions. It proves that the lower semicontinuous regularization of $F$ is closed and proper and agrees with $F$ on the relative interior of its effective domain. Since $F$ is proper, this regularization is exactly the closure defined here.

Outside the closure of $\operatorname{Dom}F$, both functions are $+\infty$: a neighborhood misses the effective domain, so the lower semicontinuous regularization is $+\infty$ there. Every remaining point outside the relative boundary belongs to the relative interior of the domain. Therefore disagreement is possible only on that relative boundary, as asserted.