The definition I know of martingale is the following: given a probability space $(\Omega,\mathcal{F},P)$, a stochastic process $\{X_t\}_{t\geq0}$ and a filtration $\{\mathcal{F}_t\}_{t\geq0}$, we say that $\{X_t\}_{t\geq0}$ is a martingale if:
$X_t\in L^1(\Omega)$ for all $t\geq0$,
$X_t$ is $\mathcal{F}_t$-measurable for all $t\geq0$,
$E[X_t|\mathcal{F}_s]=X_s$ for every $0\leq s\leq t$.
I would like to know if that definition could be extended to the case $X_t\geq0$ for all $t\geq0$, because the expectation for nonnegative random variables is well-defined (in $[0,+\infty]$).
Motivation: I read that, if $\{X_t\}_{t\geq0}$ is a martingale, then $\{|X_t|^p\}_{t\geq0}$ is a submartingale (by Jensen's inequality). However, there is no guarantee that $|X_t|^p\in L^1(\Omega)$, so I would like to know if the fact that $|X_t|^p\geq0$ avoids the problem.