Continuous square integrable martingales and family of stopping paths

Question

I have the following question and I'm not sure of the solution that I've been given:

"Show that for an element $M\in\mathcal{M}_c^2$, the family $\{M_T\}_{T\in\mathcal{J}_a}$ is uniformly integrable for any $a>0$"

By definition I have that $\mathcal{J}_a = \{\text{T a stopping time and} \mathbb{P}(T\leq a)=1\}$

The solution I have been given is very brief anf I do not understand:

$\mathbb{E}[M_a|\mathcal{F}_T]=M_T$

$\mathbb{E}[M_a^2|\mathcal{F}_T]\geq M_T^2$

Then use something like Chebyshev

I am completely confused by this answer, could somebody shed some light on it? Thank you!

Answer 1

Let us admit that $$\tag{*}\mathbb{E}[M_a|\mathcal{F}_T]=M_T.$$ Then $$ \left\lvert M_T\right\rvert\leqslant \mathbb{E}\left[\left\lvert M_a\right\rvert|\mathcal{F}_T\right] $$ and you probably now that for each non-negative integrable random variable $Y$ on a probability space $(\Omega,\mathcal F,\Pr)$, the family $\left\{\mathbb E\left[Y\mid \mathcal G\right],\mathcal G\mbox{ is a sub sigma-algebra of }\mathcal F\right\}$ is uniformly integrable.

In order to prove (*), you can try to start with the case where $T$ has finitely many values, and then look at $T_n=2^n\lfloor 2^{-n}T\rfloor$.