proving Lenglart's dominated property for some function $\textbf{Definition:}$ $X$ is $L-$dominated by $Y$ if $\mathbb{E}(|X_T|) \leq \mathbb{E}(|Y_T|)$ for every bounded stopping time $T$. Define $\mathbb{N}:=\{1,2,3,...\}$, i.e. the natural number sequence.
$\textbf{Question:}$  Suppose I have a non-negative process $X = (X_n)_{n\in \mathbb{N}}$ that is an $\mathbb{F}-$submartingale, i.e. $\mathbb{E}(X_n|\mathcal{F}_{n-1}) \geq X_{n-1}$, where $X_n$ is $\mathcal{F}_n \subset \mathbb{F}$ measurable, and $X_n$ is integrable for each $n$.
I want to show that $X$ is $L-$dominated by $Y = (Y_n)_{n\in\mathbb{N}}$, where we denote $$Y_n:=X_1+A_n,$$ $$A_n:=\sum_{j=2}^n \mathbb{E}(\Delta X_j|\mathcal{F}_{j-1}),\;\;\;A_1:=0$$ and $$\Delta X_j := X_j-X_{j-1}$$
 A: Note first that $X_n - Y_n$ is a martingale. To see this, notice $A$ is predictable, i.e. $A_n$ is $\mathcal{F}_{n-1}$ measurable for every $n$. Then we have
\begin{align*}
\mathbb{E}(X_n-Y_n|\mathcal{F}_{n-1}) &= \mathbb{E}(X_n-X_1-A_n|\mathcal{F}_{n-1}) \\
&= \mathbb{E}(X_n-\mathbb{E}(\Delta X_n|\mathcal{F}_{n-1})|\mathcal{F}_{n-1})-X_1-A_{n-1} \\
& =X_{n-1}-X_1-A_n \\
&=X_{n-1}-Y_{n-1},
\end{align*}
By Doob's Optimal Stopping Theorem, we have $$\mathbb{E}(X_T-Y_T|\mathcal{F}_1) = X_1-Y_1,$$ where $T$ is any bounded stopping time taking values in $\mathbb{N}$. Rearrangement yields
\begin{align*}
\mathbb{E}(X_T|\mathcal{F}_1) &= \mathbb{E}(Y_T|\mathcal{F}_1)+X_1-Y_1 \\
& =\mathbb{E}(Y_T|\mathcal{F}_1),
\end{align*}
implying that $$\mathbb{E}(X_T)=\mathbb{E}(Y_T).$$
Finally, as $X_n \geq 0$ by assumption and $A_n$ is increasing, i.e. $0\leq A_1 \leq A_2 \leq...$, then $$\mathbb{E}(|X_T|)=\mathbb{E}(X_T)=\mathbb{E}(Y_T)=\mathbb{E}(|Y_T|),$$
showing that $X$ is $L-$dominated by $Y$ as required to show.
