Doob's submartingale theorem According to Doob's theorem, If $\{X_n,\mathcal{F}_n, n\in \mathbb{N}^* \}$ is a submartingale and $L_1$ - bounded, it means
$$\sup\limits_{n\ge 1} \mathbb{E} (|X_n|)<\infty,$$
then $\{X_n\}$ converges almost surely to a certain random variable $X_\infty$. In my opinion,  If $\{X_n,\mathcal{F}_n, n\in \mathbb{N}^* \}$ is a submartingale then by definition, $\mathbb{E} (|X_n|)<\infty$ for all $n\in \mathbb{N}^*$, why must we add condition $\sup\limits_{n\ge 1} \mathbb{E} (|X_n|)<\infty$? I think they are equivalent. Can you explain this?

Say $X_n = n$ almost surely. Then $\mathbb{E}|X_n| <\infty$, but $\sup_{n}\mathbb{E}|X_n| = \infty$.
In other words, we need the expectations of $|X_n|$ to be uniformly bounded.
