PomeTheorem 17.10.63.D84A6072F1A6 The functional $E_{\theta }^{\phi }\colon \mathcal{E}^1(X,\theta ;\phi ) \rightarrow \mathbb {R}$ is continuous.
Theorem 17.10.63D84A6072F1A6
Proof. Let $(\varphi _ j)_{j>0}$ be a sequence in $\mathcal{E}^1(X,\theta ;\phi )$ with $d_1$-limit $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$. We wish to show that
1C0180D1496E
17.10.64
\begin{equation} \lim _{j\to \infty } E_{\theta }^{\phi }(\varphi _ j)=E_{\theta }^{\phi }(\varphi ). \end{equation}
It suffices to prove that every subsequence has a further subsequence along which (17.10.64) holds. Thus, after replacing $(\varphi _ j)_{j>0}$ by a subsequence, let $(\eta _ j)_ j$ and $(\psi _ j)_ j$ be the monotone sequences supplied by 17.10.62. From the construction, $\eta _ j\leq \varphi _ j\leq \psi _ j$. By proposition 16.5.37,
\[ E_{\theta }^{\phi }(\eta _ j)\to E_{\theta }^{\phi }(\varphi ),\quad E_{\theta }^{\phi }(\psi _ j)\to E_{\theta }^{\phi }(\varphi ). \]
The monotonicity of $E_{\theta }^{\phi }$ then squeezes $E_{\theta }^{\phi }(\varphi _ j)$, proving (17.10.64). $\square$
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