## Ito-Integral

Hallo,

I have to show that the following sets of stochastic processes are not equal:

$L^2:=\{(X_t)|(X_t)$ is progessive measurable and $\mathbb{E}[\int\limits_0^T X_t^2dt]< \infty \}$

$H^2:=\{(X_t)|(X_t)$ is progessive measurable and $\int\limits_0^T X_t^2dt< \infty$ a.s. $\}$

I know that these sets aren't equal.
But how can I show that $\int\limits_0^T X_t^2dt< \infty$ a.s. doesn't imply $\mathbb{E}[\int\limits_0^T X_t^2dt]< \infty$

Does anyone know a proof?

I also know that the Itô-integral $\int\limits_0^T X_t dW_t$ where $W_t$ is a standard Brownian motion and $X_t \in L^2$ is not just a local martingale, it's also a martingale.

But is $\int\limits_0^T X_t dW_t$ also a martingale if $X_t \in H^2$?

Can anybody help me?
Thanks!