Proof.
We first note that as a consequence of 17.10.53, $(\sup _ X \varphi _ i)_{i>0}$ is bounded. Let $C_ n$ be the constant in theorem 17.10.48. After replacing $(\varphi _ i)_{i>0}$ by a subsequence, we may assume that
\[ d_1(\varphi _ i,\varphi _{i+1})\leq \min \{ C_ n^{-2i},2^{-i}\} . \]
We first construct $(\psi _ i)_{i>0}$. For this purpose, it suffices to define
\[ \psi _ i\coloneqq \operatorname *{sup}_{j\geq i}{}\! {}^{*} \varphi _ j \]
for each $i>0$. It follows from proposition 16.5.53 that $\psi _ i\in \mathcal{E}^1(X,\theta ;\phi )$ for each $i>0$. For $k\geq 0$, set
\[ \psi _ i^ k\coloneqq \varphi _ i\lor \varphi _{i+1}\lor \cdots \lor \varphi _{i+k}. \]
Iterating corollary 17.10.51 gives
\[ d_1(\varphi _ i,\psi _ i^ k)\leq \sum _{a=i}^{i+k-1}C_ n^{a+1-i}d_1(\varphi _ a,\varphi _{a+1})\leq \sum _{a=i}^{i+k-1}C_ n^{a+1-i}C_ n^{-2a}. \]
Since $\psi _ i^ k$ increases almost everywhere to $\psi _ i$, letting $k\to \infty $ and using proposition 16.5.37, we get
\[ d_1(\varphi _ i,\psi _ i)\leq \frac{C_ n^{1-2i}}{1-C_ n^{-1}}. \]
Thus $\psi _ i\to \varphi $ in $d_1$. Since $(\psi _ i)_ i$ is decreasing, the monotone case in the proof of theorem 17.10.60 shows that $\psi _ i$ converges almost everywhere to $\varphi $, hence pointwise by proposition 5.14.6.
Next we construct $(\eta _ i)_{i>0}$. For each $i>0$ and $k\geq 0$, we let
\[ \eta _ i^ k\coloneqq \varphi _ i\land \varphi _{i+1}\land \cdots \land \varphi _{i+k}. \]
Then
\[ \begin{aligned} d_1(\varphi _ i,\eta _ i^ k)\leq & \sum _{j=0}^{k-1} d_1\left(\eta _ i^ j,\eta _ i^{j+1}\right)\\ \leq & \sum _{j=0}^{k-1} d_1\left(\varphi _{i+j},\varphi _{i+j+1}\right) & \textup{by proposition \href{../../tag/61DD221912DE/}{17.10.43}}\\ \leq & \sum _{j=0}^{k-1} 2^{-i-j}\\ \leq & 2^{1-i}. \end{aligned} \]
Since $\eta _ i^ k\leq \varphi _ i$, the preceding estimate gives
\[ E_{\theta }^{\phi }(\eta _ i^ k)\geq E_{\theta }^{\phi }(\varphi _ i)-2^{1-i},\quad k\geq 0. \]
The sequence $(\eta _ i^ k)_{k\geq 0}$ is decreasing, and the monotonicity of $E_{\theta }^{\phi }$ shows that the sequence $\left(E_{\theta }^{\phi }(\eta _ i^ k)\right)_{k\geq 0}$ is decreasing and bounded from below, hence convergent. Thus, for $k\leq l$,
\[ d_1(\eta _ i^ k,\eta _ i^ l)=E_{\theta }^{\phi }(\eta _ i^ k)-E_{\theta }^{\phi }(\eta _ i^ l), \]
and $(\eta _ i^ k)_{k\geq 0}$ is a decreasing Cauchy sequence. Therefore, the monotone case in the proof of theorem 17.10.60 shows that
\[ \eta _ i\coloneqq \lim _{k\to \infty }\eta _ i^ k=\inf _{k\geq 0} \eta _ i^ k \]
is the $d_1$-limit of $(\eta _ i^ k)_{k\geq 0}$ and proposition 16.5.37 shows that
\[ d_1(\varphi _ i,\eta _ i)\leq 2^{1-i}. \]
By construction, $\eta _ i\leq \varphi _ i$ and $(\eta _ i)_ i$ is increasing. Therefore, $\varphi $ is the $d_1$-limit of the increasing sequence $(\eta _ i)_{i>0}$. As we have seen in the proof of theorem 17.10.60, this implies that $\varphi $ is also the almost everywhere limit of $(\eta _ i)_{i>0}$.
$\square$
Pome
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