In the definition of martingales from Wikipedia there are two conditions. It must be an adapted and integrable process for all $n \in \mathbb{N}$.
My question is just: What could happen if we omit the condition that $\mathbb{E}(|X_n|) < \infty$ for all $n \in \mathbb{N}$ and we allow it to be $\infty$?
My Thoughts: Since the operator $\mathbb{E}: \chi \to \mathbb{R}$ is linear, where $\chi\subset L^1(\Omega)$ is the space of random variables. Maybe dropping the second condition could lead to this operator not to be bounded.
Many thanks