Pome2FF0E77E9188Proposition 17.4.4.2FF0E77E9188 The set $\{ u\in \mathrm{PSH}(X,\theta ):u\preceq \phi \} $ is closed in $L^1(X)$, and $E_\theta ^\phi $ is upper semicontinuous on it, with values in $[-\infty ,\infty )$: if $u_ j\to u$ in $L^1(X)$, then
Proof. Let $u_ j\preceq \phi $ converge to $u$ in $L^1(X)$. By lemma 17.4.3, the $u_ j$ are uniformly bounded above, so lemma 16.22.2 gives a constant $C$ with $u_ j\leq \phi +C$ for all $j$. Since $\phi +C$ is upper semicontinuous, the same lemma gives $u\leq \phi +C$, which proves closedness. The lemma also shows that the tail envelopes $v_ j=(\sup _{k\geq j}u_ k)^*$ are $\theta $-psh potentials which decrease pointwise to $u$; moreover $v_ j\leq \phi +C$, because $\phi +C$ is upper semicontinuous and dominates every $u_ k$. Since $u_ j\leq v_ j$, monotonicity and proposition 17.3.13 give
Since $L^1(X)$ is a metric space, sequential upper semicontinuity is upper semicontinuity. $\square$
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