So we had the following example in class:
Assume $(\Omega, ℱ,(ℱ_{t}),ℙ)$ a filtered probability space and $X: \Omega \to \mathbb{R}$ a random variable such that $$ \mathbb{E}[|X|^{2}] < \infty \quad . $$
Then the stochastic process $(M)_{t\geq 0}$ defined by
$$ M_t := \mathbb{E}[X|ℱ_{t}] $$
for all $t \geq 0$ is a $ℱ_{t}$ - martingale.
So I was wondering whether in this case the martingale $(M)_t$ is square-integrable, i. e.
$$ \mathbb{E}[|M_t|^{2}] < \infty $$
for all $t \geq 0$, because of the square integrability of the random variable $X$.
Many thanks in advance.