Counterexample for Interchange of Limits in integration

Question

If $f_n$ converges to $f$ uniformly in $\mathbb{R}$, then

\begin{equation*} \lim_{n\to\infty}\int_a^b f_n(x)\,dx =\int_a^b f(x)\,dx \end{equation*}

but it's not true in general that

\begin{equation*} \lim_{n\to\infty }\int_{-\infty}^\infty f_n(x)\,dx =\int_{-\infty}^\infty f(x) \, dx. \end{equation*}

I cannot think of any counterexample.

Answer 1

Take $f_n = {1 \over 2n} 1_{[-n,n]}$, then $f_n \to 0$ uniformly, but $\int f_n = 1, \int f = 0$.