Proposition 21.4.4374D75E96CF6

Proof. Let $A_ j=\max \{ F,h_ P-j\} $. These are convex functions, decreasing in $j$, with $A_ j\sim h_ P$, and $A_ j=F$ on every compact set on which $h_ P-F\leq j$; hence $A_ j\to F$ locally uniformly. With the mollifiers $\rho _\varepsilon $ of lemma 21.4.3, choose $\varepsilon _ j\downarrow 0$ and set $F_ j=A_ j*\rho _{\varepsilon _ j}$. By that lemma, these functions are smooth and convex, and the monotonicity in $\varepsilon $ together with $A_{j+1}\leq A_ j$ gives $F_{j+1}=A_{j+1}*\rho _{\varepsilon _{j+1}}\leq A_ j*\rho _{\varepsilon _{j+1}}\leq A_ j*\rho _{\varepsilon _ j}=F_ j$, and $F_ j\geq A_ j\geq F$. By proposition 21.4.2, $F$ and $h_ P$ are Lipschitz with constant $\max _{p\in P}|p|$, hence so is $A_ j$, and therefore $|F_ j-A_ j|\leq C\varepsilon _ j$ with $C=\max _{p\in P}|p|\int |y|\rho (y)\, \mathrm dy$. Consequently $F_ j\sim A_ j\sim h_ P$, so $F_ j\in \mathcal E^\infty (N_{\mathbb R},P)$, and $F_ j\to F$ locally uniformly.

If $F_ j\downarrow F$ pointwise, then

and the conjugates $F_ j^*$ increase, because conjugation reverses inequalities. Finally, if a sequence of finite convex functions on $N_{\mathbb R}$ converges pointwise to a finite function, it converges locally uniformly by proposition 21.2.6, applied with $\Omega $ and the dense subset both equal to $N_{\mathbb R}$, and conversely local uniform convergence implies pointwise convergence. $\square$