Lemma 5.41.8FAE98D3A3F69

Proof. Fix $\tau <c$ and let $\tau _ j\uparrow \tau $. The decreasing limit of the functions $\Gamma _{\tau _ j}$ dominates $\Gamma _\tau $, so it belongs to $\mathrm{PSH}(X,\theta )$. Choose $r\in (\tau ,c)$. Outside the pole set of $\Gamma _ r$, the concave function $s\mapsto \Gamma _ s(x)$ is finite near $\tau $ and hence continuous there. Thus the decreasing limit equals $\Gamma _\tau $ almost everywhere, and hence everywhere by lemma 5.24.3. This proves left continuity at every $\tau <c$; at $c$ it holds by definition. Since the function is decreasing, left continuity gives upper semicontinuity, also where the value is $-\infty $, and concavity survives the extension. The inverse formula is lemma 5.41.5 at each $x$. $\square$