Convergence in $L^2( [0,T] \times \Omega )$ implies uniform convergence

Question

Let $X^n$ be a family of continuous stochastic processes such that $E[ \sup_{t \in [0,T} \mid X^n_t \mid ^2 ] < \infty $ for all $n.$

We assume that $$ \lim_{n \to \infty} E\left[ \int_0^T \mid X^n_t \mid^2 dt \right] = 0. $$

Doe this imply $$ \lim_{n \to \infty} E\left[ \sup_{t \in [0,T] } \mid X^n_t \mid^2 \right] = 0 \quad ? $$

Answer 1

No. Take $X_t^{n}(\omega) =t^{n}$ for all $n,t,\omega$ with $T=1$.