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Proposition 5.14.201C3654494E2

Proposition 5.14.2.01C3654494E2

  1. 8BD4DEED7DA3

    Assume that $(\varphi _ i)_{i\in I}$ is a non-empty family in $\mathrm{PSH}(X)$ that is locally uniformly bounded from above. Then $\operatorname *{sup}_{i\in I}{}\! {}^{*} \varphi _ i\in \mathrm{PSH}(X)$.

  2. 567577F8D619

    Assume that $(\varphi _ i)_{i\in I}$ is a decreasing net in $\mathrm{PSH}(X)$ such that $\varphi \coloneqq \inf _{i\in I}\varphi _ i$ is not identically $-\infty $ on any connected component of $X$. Then $\varphi \in \mathrm{PSH}(X)$. Furthermore, $\varphi _ i\xrightarrow {L^1_{\mathrm{loc}}} \varphi $.

Proof. For the upper envelope, the upper-envelope theorem applies exactly to the given nonempty, locally uniformly upper-bounded family; its conclusion is that the upper semicontinuous regularization of its supremum is plurisubharmonic. For a decreasing net, the decreasing-net theorem gives, on each connected component, either a plurisubharmonic limit or the limit identically $-\infty$, with the local convergence asserted there. Applying these results in charts and using uniqueness of representatives on overlaps proves the statements on a manifold. No passage from an arbitrary net to a supposedly cofinal sequence is needed.

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