Examples of a uniformly convergent sequence of functions with the given property

Question

I am trying to find a sequence $(f_n)$ of continuous functions on $\mathbb{R}$ such that $f_n\to f$ uniformly on $\mathbb{R}$, but $$\lim_{n\to \infty} \int_{-\infty}^{\infty}f_n(x)\ne\int_{-\infty}^{\infty}f(x).$$ Please help.

Answer 1

Consider the sequence of continuous functions $(f_n)_{n\geq 1}$, where $$f_n(x):=\begin{cases} 1/n &\mbox{if $|x|\leq n$}\\ \frac{(n+1)-|x|}{n} &\mbox{if $n\leq |x|\leq n+1$}\\ 0 &\mbox{if $|x|\geq n+1$}.\\ \end{cases}$$ which converges to $0$ uniformly in $\mathbb{R}$. Then $$\int_{-\infty}^{\infty}f_n(x)\,dx\ge 2>0=\int_{-\infty}^{\infty}f(x)\, dx.$$

Answer 2

The sequence defined as

$$f(x) = \begin{cases} 1/n, & \mbox{if } n\mbox{ 0 $\le x \le n$ } \\ 0, & \mbox{if } \mbox{$x>n $ and $x<0$} \end{cases}$$

Is not a sequence of continous function but has the property

$$1=\lim_{n\to \infty} \int_{-\infty}^{\infty}f_n(x)\ne\int_{-\infty}^{\infty}f(x)=0$$

A little change of this sequence gives you what you want.