# Problem about almost sure and L1 convergence

1. Nov 11, 2009

### yavanna

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

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

• Proove that $$Y<\infty$$ a.s.
• Let $$Q$$ a new probability measure defined as it follows:
$$\displaystyle Q(A)=\frac{1}{c} \mathbb{E}\!\left[1_{A} \frac{1}{1+Y}\right]$$, where $$\displaystyle c=\mathbb{E}\!\left[\frac{1}{1+Y}\right]$$.
Proove that $$X_{n} \rightarrow X$$ (in $$L_{1}(Q)$$).

Last edited: Nov 11, 2009