According to: A German lecture on mathematical finance (page 9)
In the context of the lecture $(X_t)_{t=0,\ldots,T} \in \mathbb{R}^d$ is a stochastic process, that models a finance market, where $X^1_t = 1$ and $X_t$ is adapted to the Filtration $\mathcal{F}$ ($X_t$ is $\mathcal{F}_t$ measurable).
A self-financed portfolio $(H^i_t)\in\mathbb{R}^d$ is a stochastic process, that satisfies: $H_t$ is $\mathcal{F}_{t-1}$-measurable & $H_t \cdot X_t = H_t \cdot X_{t+1}$
The Doob's System Theorem states: The following statements are equivalent:
$(i)$ $(X_t)$ is a martingale w.r.t. $\mathcal{F}_t$
$(ii)$ for all bounded self-financed portfolios $H$ we have: $v_t(H)=H_t \cdot X_t$ is a martingale w.r.t. $\mathcal{F}_t$
$(iii)$ for all self-financed portfolios $H$, that satisfy $\mathbb{E}(v_t(H)_{-})<\infty$ (where $f(x)_{-}:=max\{0,-f(x)\}$), we have: $v_t(H)$ is a martingale w.r.t. $\mathcal{F}_t$
$(iv)$ for all self-financed portfolios $H$, that satisfy $v_t(H)\geq 0$ almost surely, we have: $\mathbb{E}(v_T(H))=v_0(H)$
Somehow I can't find this theorem somewhere in the internet or on this site. (Maybe it's name isn't correctly sated?) The equivalence of $(i)\Leftrightarrow(ii)$ are pretty easy to show, as well as the implications $(iii)\Rightarrow(i)$ and $(iii)\Rightarrow(iv)$. But how does $(i)$ or $(ii)$ implicate $(iii)$? How do we prove, that $v_t(H)\in L^1$? And how can we show $(iv){\Rightarrow} (i),(ii) \text{ or } (iii)$ ?
Any help or advice would be greatly appreciated. Thank you.