I am trying to prove the convergence of the function $f\_n = I\_{[n,n+1]}$ to $f=0$, but first of all I don't in which way it converges, either in $\mathcal{L}\_p$-measure or stochastically, or maybe some other form of convergence often used in measure-theory.

For now I'm assuming it's stochastic convergence, as in the following:

$$ \text{lim}\_{n \rightarrow \infty} \, \mu(\{x \in \Re: |f\_n(x)-f(x)| \geq \alpha\}\cap A )=0$$

must hold for all $\alpha \in \Re\_{>0}$ and all $A \in \mathcal{B}(\Re)$ of finite measure.

I know it must be true since there is no finite $A$ for which this holds. Could someone give me a hint how to start off this proof?