A sequence of continuous functions

Question

There exist a sequence of continuous functions $\{f_n\}$ on $\mathbb{R}$ such that $\{f_n\}$ converges to $f$ uniformly on $\mathbb{R}$, but

$$\lim_{n\rightarrow\infty}\int_{-\infty}^{\infty}f_n(x)dx \neq \int_{-\infty}^{\infty}f(x)dx$$

solution I tried-

as we know that if sequence of continuous function $f_n$ converges to a fucntion $f$ over a given interval $[a,\infty)$ then $$\lim_{n\rightarrow\infty}\int_{a}^{\infty}f_n(x)dx = \int_{a }^{\infty}f(x)dx$$ so in the above question they replace the $a$ with $-\infty$ does it effects the result?

Chinnapparaj R · Accepted Answer · 2019-06-12T15:41:41.267

0

Your method is not correct. Actually, for this one, you find a counterexample for the existence.

Consider $$f_n(x)=\frac{1}{n(1+\vert x \vert)}$$ Now try to complete the details!

edited Jun 12 '19 at 15:41

answered Jun 12 '19 at 15:37

Chinnapparaj R

11,589

does it only work when interval is compact? – TheStudent Jun 12 '19 at 15:41
The indefinite integral is divergent here – Chinnapparaj R Jun 12 '19 at 15:47
1

Oh...you mean ..about original problem? Yes, for compact intervals , limit of the integral equals integral of the limit – Chinnapparaj R Jun 12 '19 at 15:54

A sequence of continuous functions

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