Proposition 16.5.2037860FDCEAB5

Proof. We first handle the case where $I=\mathbb {Z}_{>0}$. Namely, $(\varphi _ i)_ i$ is a decreasing sequence. Thanks to 16.5.14, we know that $E_{\theta }^{\phi }(\varphi _ i)$ is decreasing in $i$. So the limit in (16.5.21) exists and the $\geq $ inequality holds. Conversely, let $\psi \in \mathcal{E}^{\infty }(X,\theta ;\phi )$ and $\varphi \leq \psi $. We need to show that

Replacing the sequence $(\varphi _ i)_ i$ by $(\varphi _ i\lor \psi )_ i$, we are reduced to proving the $\leq $ direction of (16.5.21) when $\varphi \in \mathcal{E}^{\infty }(X,\theta ;\phi )$.

In this case, thanks to 16.5.16, we have

Our assertion follows from the monotone convergence theorem.

We now treat the general net case. By 16.5.14, the net $E_{\theta }^{\phi }(\varphi _ i)$ is decreasing and bounded below by $E_{\theta }^{\phi }(\varphi )$. Thus

exists and satisfies $L\geq E_{\theta }^{\phi }(\varphi )$. It remains to prove the reverse inequality.

Since $\varphi \leq \varphi _ i$ for every $i\in I$, the hypotheses of corollary 5.17.5 are satisfied, and hence $\varphi _ i\xrightarrow {L^1}\varphi $. For each $k\geq 1$, choose $a_ k\in I$ such that

We can recursively choose an increasing sequence $(i_ k)_{k\geq 1}$ in $I$ such that $i_ k\geq a_ k$ for all $k$. Then $(\varphi _{i_ k})_ k$ is a decreasing sequence and $\varphi _{i_ k}\to \varphi $ in $L^1$. Its pointwise decreasing limit is therefore equal to $\varphi $ almost everywhere, hence everywhere by proposition 5.14.6. Applying the sequence case to $(\varphi _{i_ k})_ k$, we get

Since $L\leq E_{\theta }^{\phi }(\varphi _{i_ k})$ for every $k$, we conclude $L\leq E_{\theta }^{\phi }(\varphi )$. $\square$