Sequence of Lebesgue integrable functions on R that converges pointwise but for which term by term integration is not valid

Question

Can anyone give an example of a sequence of Lebesgue integrable functions on R that converges pointwise but for which term by term integration is not valid?

I know that the Lebesgue Monotone Convergence Theorem does not apply to such an example but I cannot think of an example to s how this.

Answer 1

Maybe you mean something like this?: let $f_n(x):=\chi_{[n,n+1]}(x)$ for $n\in\Bbb N$. Then clearly $f_n\in\mathcal L_1(\Bbb R)$ and $(f_n)\to 0$ point-wise, but

$$\lim_{n\to\infty}\int_{\Bbb R} f_n(x)\,\lambda(dx)=1\neq\int_{\Bbb R} \lim_{n\to\infty}f_n(x)\,\lambda(dx)=0$$

Answer 2

Take $f_n (t) = n^{-1} ,$ for $t\in [0, n] $ and $f_n (t) =0$ othervise, then $f_n \to 0 $ uniformly , but the equality $$\lim_{n\to\infty }\int_{\mathbb{R} } f_n (t) dt =\int_{\mathbb{R}}\lim_{n\to\infty } f_n (t) dt $$ does not hold true.