Let $M:\Omega\times\mathbb{R^+}\longrightarrow \mathbb{R}$ be a martingale such that \begin{equation*} \sup_{t\in \mathbb{R}^+}\mathbb{E}[M_t^2]<\infty \end{equation*} Then is $M$ uniformly integrable? i.e. is it true that \begin{equation*} \lim_{K\rightarrow\infty} \left[ \sup_{t\in \mathbb{R}^+} \mathbb{E}\left[ M_t^2 \mathbb{I}_{\left\{M_t^2> K\right\}}\right]\right] =0 \end{equation*} I think it is...and my attempt at proving it so far is as follows:
However I am getting stuck at the last step where I need to interchange the limit and the supremum...(I have highlighted this step in red). I know that we can interchange the limit and the suprermum if $\sup_{t\in \mathbb{R}^+} \mathbb{E}\left[ M_t^2 \mathbb{I}_{\left\{M_t^2> K\right\}}\right]$ converges uniformly to 0 when we take the limit as $C\rightarrow\infty$, but I couldn't show this either....
Any help would be greatly appreciated, whether it be: 1) showing another way of proving it, 2) showing a way of getting around/proving the last step 3) telling me this statement isn't true (and if it isn't is there a counter example).
Many thanks!