Proposition 21.11.29C3CC440747D6

Proof. Put $H=\operatorname{cl}\operatorname{CE}F$. Since $H\leq F$, order reversal gives $H^*\geq F^*$. Fix $p$ with $F^*(p)$ finite. The affine function $\ell(x)=\langle p,x\rangle-F^*(p)$ lies below $F$, hence below its convex envelope and below its lower semicontinuous regularization, because $\ell$ itself is continuous. Thus $\langle p,x\rangle-H(x)\leq F^*(p)$ for all $x$, proving $H^*(p)\leq F^*(p)$.

If $F^*(p)=+\infty$, the reverse inequality already gives equality. If $F^*(p)=-\infty$, then $F$ is identically $+\infty$; its closed convex envelope is also identically $+\infty$, and the equality is immediate. This treats every possible value of the conjugate.