In a book I'm reading (but I get the impression that many others proceed similarly), when proving that a stochastic process is a martingale, the author proceeds as if integrability is trivial, or somehow given, and simply focus on proving the part of the definition involving the conditional expectation.
For example: Let $M_t = B_t^2-t$, where $B_t$ is a Brownian motion. He simply focus on proving that $E(M_t\mid \mathcal{F_s})=M_s$ for $s<t$. How does the author know that $E(|B_t^2-t| )<\infty$?
Is there a theorem/property that makes it easy to ensure that the processes are integrable?