Can anyone give me some advice about this problem? Thanks.

Let $\displaystyle \lim_{n \to \infty}X_{n}=X$ a.s.

And let $\displaystyle Y=\sup_n|X_{n}-X|$.

- Proove that $\displaystyle Y<\infty$ a.s.
- Let $\displaystyle Q$ a new probability measure defined as it follows:

$\displaystyle \displaystyle Q(A)=\frac{1}{c} \mathbb{E}\!\left[1_{A} \frac{1}{1+Y}\right]$, where $\displaystyle \displaystyle c=\mathbb{E}\!\left[\frac{1}{1+Y}\right]$.

Proove that $\displaystyle X_{n} \rightarrow X$ (in $\displaystyle L_{1}(Q)$).