Let $F_n$ converge to $F$ uniformly on all of $\Bbb{R}$, show that $\int F_n$ need not converge to $\int F$.

Question

Let $F_n$ converge to $F$ uniformly on all of $\Bbb{R}$, show that $\int F_n$ need not converge to $\int F$.

I know that this is true when $F_n$ converges on $[a,b]$.

I believe that it fails if the interval is all $\Bbb{R}$. Please explain.

Answer 1

You could have a look at the following sequence of functions:

\begin{equation} F_n(x)=\begin{cases} \frac{1}{n} \qquad x\in [0,n],\\ 0\qquad else \end{cases} \end{equation}

Answer 2

$(1/n)\chi[0,n]$ converges uniformly to 0 but $\int f_n(x)$=1 for any fixed n