Suppose $f$ is integrable on sets $A\_1 \supset A\_2 \supset \ldots \supset A\_n \supset \ldots$ and let $A = \cap\_{n=1}^{\infty} A\_n$.

Does $\int\_{A\_n} f(x) d\mu$ converge to $\int\_{A } f(x) d\mu$?

I think this is false but I can't prove it, If you can give me some hints please, I'm really stuck in this problem.

Thanks for your time and help.