Proposition 17.4.1049F75F5547D

Proof. By lemma 16.22.2, $u_ i\leq \phi +\sup _ Xu_ i$, and the suprema $\sup _ Xu_ i$ are bounded above by hypothesis; hence $u\preceq \phi $. Since $u$ dominates every $u_ i$, monotonicity of the energy and theorem 17.3.14 show that $u\in \mathcal E^1(X,\theta ,\phi )$.

Bounded differences. Consider first an increasing sequence $(u_ j)$ and a constant $C\geq 1$ such that $\phi -C\leq u_ j\leq u\leq \phi $ for all $j$. The pole sets of $u_ j$ and $u$ coincide with that of $\phi $. Let $f_ j$ be the function equal to $u-u_ j$ outside this pole set and to $0$ on it; it is a Borel function with values in $[0,C]$. By the tangent bound (17.3.10),

and we show that the right side tends to zero by applying lemma 17.2.2 with $\nu _ j=\mu _{u_ j}$, $\nu =\mu _ u$, the weights $f_ j$ and the limit weight $0$. For each $j$ the potential $C^{-1}u_ j+(1-C^{-1})\phi $ lies between $\phi -1$ and $\phi $ and its measure is at least $C^{-n}\mu _{u_ j}$; hence $\mu _{u_ j}\leq C^ n\operatorname {Cap}_{\theta ,\phi }$ on Borel sets, and likewise $\mu _ u\leq C^ n\operatorname {Cap}_{\theta ,\phi }$. By theorem 16.27.17, the measures $\mu _{u_ j}$ and $\mu _ u$ are therefore uniformly absolutely continuous with respect to $\operatorname {Cap}_\omega $. They converge weakly by corollary 13.8.9, since $u_ j$ increases to $u$ quasi-everywhere (corollary 5.47.11).

It remains to show that $f_ j\to 0$ in capacity. Fix finitely many pairs of concentric coordinate balls $B_\alpha \Subset B'_\alpha $ as in proposition 16.27.18, chosen so that $\theta $ has a smooth potential $h_\alpha $ on a neighborhood of $\overline{B'_\alpha }$ (this is possible by lemma 6.14.2, applied on slightly larger concentric balls), and let $c$ be the constant of that proposition. Let $\delta ,\eta >0$. By proposition 16.27.18(3) there is an open set $G$ with $\operatorname {Cap}_\omega (G)<\eta $ which contains the pole set of $\phi $ and such that $\phi $ is finite and continuous on the compact set $X\setminus G$. Let $L$ be a common bound for $|\phi |$ on $X\setminus G$ and for the functions $|h_\alpha |$ on $B'_\alpha $. The functions

are bounded plurisubharmonic functions on $B'_\alpha $, and $w_{\alpha ,j}$ increases to $w_\alpha $ almost everywhere, because $u_ j$ increases to $u$ quasi-everywhere and pluripolar sets have Lebesgue measure zero. By theorem 7.6.1, $w_{\alpha ,j}\to w_\alpha $ locally in capacity on $B'_\alpha $. On $B_\alpha \setminus G$ we have $u_ j+h_\alpha \geq \phi -C-L\geq -2L-C$, so the truncations are inactive there and $w_\alpha -w_{\alpha ,j}=u-u_ j$. Since the balls $B_\alpha $ cover $X$,

By the countable subadditivity of $\operatorname {Cap}_\omega $ (proposition 16.25.6) and the bound $\operatorname {Cap}_\omega (E)\leq c\operatorname {Cap}_{B'_\alpha }(E)$ for Borel sets $E\subset B'_\alpha $ in proposition 16.27.18,

where we also used that the capacity of a Borel set is at most its outer capacity. By definition 7.5.5, the sum tends to zero as $j\to \infty $. Hence $\varlimsup _ j\operatorname {Cap}_\omega (\{ f_ j>\delta \} )\leq \eta $, and since $\eta $ is arbitrary, $f_ j\to 0$ in capacity. Lemma 17.2.2 now gives $\int _ Xf_ j\, \mu _{u_ j}\to 0$, and hence $E_\theta ^\phi (u_ j)\to E_\theta ^\phi (u)$.

Sequences. Now let $(u_ j)$ be an arbitrary increasing sequence; subtracting a common constant, we may assume $u_ j\leq u\leq \phi $. Put $u_{j,k}=\max \{ u_ j,\phi -k\} $ and $u_ k=\max \{ u,\phi -k\} $. For fixed $k$, the sequence $u_{j,k}$ increases to $\max \{ \sup _ ju_ j,\phi -k\} $, whose upper semicontinuous regularization is $\max \{ (\sup _ ju_ j)^*,\phi -k\} =u_ k$, because $\phi -k$ is upper semicontinuous; moreover $\phi -k\leq u_{j,k}\leq u_ k\leq \phi $. The preceding case, with $\max \{ k,1\} $ in place of $C$, gives $E_\theta ^\phi (u_{j,k})\to E_\theta ^\phi (u_ k)$ as $j\to \infty $. We bound the truncation error uniformly in $j$. The tangent bound of proposition 17.3.19 gives

For $t\geq k$ put $q_ t=(u_ j+\phi -t)/2$. On $\{ u_ j<\phi -t\} $ we have $u_1\leq u_ j<q_ t$, and $q_ t\leq \phi -t/2$ on $X$. Since $\mu _{q_ t}\geq 2^{-n}\mu _{u_ j}$ by multilinearity, and $q_ t,u_1\in \mathcal E(X,\theta ,\phi )$, the comparison principle of proposition 16.25.4 gives

Consequently

as $k\to \infty $, uniformly in $j$; the limit is zero by dominated convergence, since $\phi -u_1$ is $\mu _{u_1}$-integrable. Since $E_\theta ^\phi (u_ k)\geq E_\theta ^\phi (u)$, letting first $j\to \infty $ and then $k\to \infty $ gives $\lim _ jE_\theta ^\phi (u_ j)\geq E_\theta ^\phi (u)$, and monotonicity gives the reverse inequality.

Nets. Let $(u_ i)_{i\in I}$ be a net as in the statement. Cover $X$ by finitely many coordinate charts $D_\beta $ on which $\theta $ has smooth potentials $h_\beta $. By theorem 5.10.2, applied to the functions $u_ i+h_\beta $ on $D_\beta $, there are countable sets $J_\beta \subset I$ such that $(\sup _{i\in J_\beta }u_ i)^*=u$ on $D_\beta $. The countable set $J=\bigcup _\beta J_\beta $ then satisfies $(\sup _{i\in J}u_ i)^*=u$ on $X$, since on each $D_\beta $ this envelope lies between the envelopes over $J_\beta $ and over $I$. Since the index set is directed, we can choose indices $i_1\leq i_2\leq \cdots $ such that $i_ k$ dominates the first $k$ elements of an enumeration of $J$. Since the net is increasing, $\sup _ ku_{i_ k}\geq \sup _{i\in J}u_ i$, so the increasing sequence $(u_{i_ k})$ has regularized upper envelope $u$, and the sequence case gives $E_\theta ^\phi (u_{i_ k})\to E_\theta ^\phi (u)$. Each of these energies is at most $\sup _ iE_\theta ^\phi (u_ i)$, which in turn is at most $E_\theta ^\phi (u)$ by monotonicity. Hence $\sup _ iE_\theta ^\phi (u_ i)=E_\theta ^\phi (u)$, and since the net of energies is increasing, its limit is this supremum. (The selected indices need not be cofinal in $I$, which is why we compare suprema.) $\square$