The definition of terms in Doob's decomposition theorem for submartingales !enter image description here The definition of $Z_n$ in the Doob's Decomposition Theorem, I think it is a predictable submartingale starting at $0$. Is that right? Thanks for your help!:)

It follows straight from the definition that any adapted increasing (integrable) process is a submartingale. This means that $Z_n \leq Z_{n+1}$ and $\mathbb{E}Z_n<\infty$ implies indeed that $(Z_n,\mathcal{F}_n)_{n \geq 0}$ is a submartingale.