21.6 Compact convex envelopes and full-mass limits2F45C4BAC276

Proof. Each $G_ i$ is bounded below: the increasing open sets $\{ G_ i>-k\} $, $k\in \mathbb N$, cover the compact set $K$, because $G_ i$ is lower semicontinuous and does not take the value $-\infty $, so one of them contains $K$; and $G_ i=+\infty $ outside $K$. Let $B$ be a common lower bound. In a combination $\sum _ j\lambda _ jG_{i_ j}(x_ j)$ as in proposition 21.3.3(2), the terms with the same function $G_ i$ can be merged: if $\Lambda _ i$ is the sum of their weights, the convexity of $G_ i$ gives $\sum _{j:i_ j=i}\lambda _ jG_ i(x_ j)\geq \Lambda _ iG_ i(y_ i)$ with $y_ i=\Lambda _ i^{-1}\sum _{j:i_ j=i}\lambda _ jx_ j$, and the barycenter is unchanged. Hence it suffices to consider combinations $\sum _{\lambda _ i>0}\lambda _ iG_ i(x_ i)$ with one point $x_ i\in \operatorname {dom}G_ i$ for each $i$ and weights $\lambda _ i\geq 0$ summing to one; for an index $i$ with $\lambda _ i=0$ we let $x_ i$ be a fixed point of $\operatorname {dom}G_ i$, and the corresponding term is omitted.

Let $x_\nu \to x$ with $\varliminf _\nu H(x_\nu )=L<+\infty $. Passing to a subsequence, we may assume that $H(x_\nu )\to L$, and we choose combinations for $x_\nu $, with points $x_{i,\nu }$ and weights $\lambda _{i,\nu }$, whose values tend to $L$. Since $K$ and the simplex of weights are compact, a further subsequence satisfies $x_{i,\nu }\to x_ i\in K$ and $\lambda _{i,\nu }\to \lambda _ i$ for every $i$. The limiting weights are nonnegative and sum to one, and since the $x_{i,\nu }$ are bounded, the terms with $\lambda _ i=0$ disappear in the limit of the barycenters, so $\sum _{\lambda _ i>0}\lambda _ ix_ i=x$. The nonnegative numbers $\lambda _{i,\nu }(G_ i(x_{i,\nu })-B)$ have sum tending to $L-B$. For $\lambda _ i>0$, lower semicontinuity gives $\varliminf _\nu \lambda _{i,\nu }(G_ i(x_{i,\nu })-B)\geq \lambda _ i(G_ i(x_ i)-B)$, and therefore

In particular, all the values $G_ i(x_ i)$ with $\lambda _ i>0$ are finite, and the combination with points $x_ i$ and weights $\lambda _ i>0$ is admissible for $x$, with value at most $L$. Thus $H(x)\leq L$, which proves that $H$ is lower semicontinuous. Moreover $B\leq H\leq G_1$, so $H$ is proper. Applying the same argument to the constant sequence $x_\nu =x$ and admissible combinations whose values tend to $H(x)<+\infty $, we see that the infimum defining $H(x)$ is attained. By proposition 21.3.3(2), $H^*=\max _ iG_ i^*$, and since $H$ is closed and proper, theorem 21.3.2 gives $H=H^{**}=(\max _ iG_ i^*)^*$.

For the last assertion, the conjugates $F_ i^*$ are closed proper convex functions whose effective domains lie in the compact set $P$, by theorem 21.3.2, proposition 21.4.2. Since $F_ i^{**}=F_ i$, the first part gives the formula $(\max _ iF_ i)^*$ for their convex envelope. $\square$

Proof. Let $\varepsilon >0$. By inner regularity there is a compact set $S$ with $\mu (W\setminus S)<\varepsilon $; let $\eta $ be a continuous function with compact support, $0\leq \eta \leq 1$ and $\eta =1$ on $S$. Since $\int \eta \, \mathrm d\mu _ j\to \int \eta \, \mathrm d\mu $ and the total masses converge, we have

so that $\mu _ j(W\setminus \operatorname{Supp}\eta )\leq \int (1-\eta )\, \mathrm d\mu _ j<\varepsilon $ for all large $j$. Thus the sequence $(\mu _ j)$ is eventually uniformly tight, and proposition 2.6.3 gives narrow convergence. (If $W$ is a point, every function on $W$ has compact support, and vague and narrow convergence coincide.) For the real Monge–Ampère measures, theorem 21.5.2 gives vague convergence and the common total mass $n!\operatorname {vol}(P)$. $\square$