Pome

Corollary 21.12.42E80DDB8667C8

Corollary 21.12.42.E80DDB8667C8 Let $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )$. Then the following are equivalent:

  1. 8FB62A15E0CB

    $\varphi \in \mathcal{E}^1(X,\omega )$;

  2. 055DBC443927

    $F_{\varphi }\in \mathcal{E}^1(N_{\mathbb {R}},P)$;

  3. 97AB544270A6

    $G_{\varphi }\in L^1(P)$.

Proof. The equivalence between 1 and 3 follows from proposition 21.12.39. The equivalence between 2 and 3 is the definition of $\mathcal{E}^1(N_{\mathbb {R}},P)$, since $F_{\varphi }^*=G_{\varphi }$. $\square$

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