$L^2$ martingales relation to martingales?

Question

Are all continuous time $L^2$-martingales i.e martingales such that $\mathbb{E}X_{t}^2<\infty$ for all $t$ also martingales in the sense that $\mathbb{E}\mid X_{t} \mid<\infty$ for all $t$, which is the common definition

Answer 1

Yes, they are. Martingales have finite first moment and $L^{2}$ martingales are martingales with the extra condition that second moments are also finite.

Answer 2

Yes, $L_2 \subset L_1$ as $$|X|<1+|X|^2.$$