Convergence pointwise but not in measure

Question

Let $X=\mathbb R$ and $\mu=m$. Let $f_n(x)=e^{-|x-n|}$ and $f(x)=0$, $x\in\mathbb R$. Show that $f_n$ converges pointwise to $f$, but $f_n$ does not converge in measure to $f$.

I didn't have any trouble showing $f_n$ converges pointwise, but my idea for convergence in measure may be flawed. It relies on the fact that $m((-\infty,-\infty)) \neq 0$, which I'm not sure is true.

Answer 1

Hint.

You need to compute $\int_{-\infty}^{+\infty} f_n(x) dx$.