Corollary 21.13.541D4A688AAB93

Corollary 21.13.54.1D4A688AAB93 There is a canonical bijection between the following sets:

1. 0180BB40C5AC

   The set of $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta ;\phi )$,

2. C7ECD31C42B2

   the set of $F\in \mathcal{P}(N_{\mathbb {R}},\Delta (\theta ,\phi ))$, and

3. 5661D01495BD

   the set of closed proper convex functions $G\in \mathrm{Conv}(M_{\mathbb {R}})$ satisfying

   \[ G|_{M_{\mathbb {R}}\setminus \Delta (\theta ,\phi )}=\infty . \]

Moreover, under these correspondences, we have the following bijections:

1. FF6FF0F617A6

   The set $\mathcal{E}_{\mathrm{tor}}(X,\theta ;\phi )$,

2. 2A1D9F970B23

   the set of $F\in \mathcal{E}(N_{\mathbb {R}},\Delta (\theta ,\phi ))$, and

3. B93778C6C5D3

   the set of closed proper convex functions $G\in \mathrm{Conv}(M_{\mathbb {R}})$ satisfying

   \[ \overline{\operatorname{Int}\{ G<\infty \} }=\Delta (\theta ,\phi ). \]