Almost uniform convergence on an open set, pointwise convergence on a closed set and an equality of a limit of an integral of the sequence to examine.

Question

$f_n: [a,b] \rightarrow \mathbb{R}$ is a function series of continous functions almost uniformly convergent on $(a,b)$ and convergent pointwise on $[a,b]$. Examine if $$\lim_{n\rightarrow \infty}\int_a^b f_n(x)\, dx = \int_a^b \lim_{n\rightarrow \infty} f_n(x)\, dx .$$ I've been wondering how to use the pointwise convergence and continuity. I do not know if such conditions preserve uniform convergence on the entire $[a,b]$.

My attempt to prove it occurs, was about defining a sequence of partitions, where, we know, there is uniform convergence on every subinterval of the partition except the two external and somehow bound the product of their suprema/infima and the subinterval lenght by defining such partition sequence. But this way I am not aware of the way for using either the pointwise convergence or the continuity.

Answer 1

For $a=0$ and $b=1$ let $f_n(x)=(n+1)x^n(1-x^n).$ Then $f_n$ is convergent uniformly to $0$ on any interval $[0,1-\delta]$ as $$0\le f_n(x)\le (n+1)(1-\delta)^n,\qquad 0\le x\le 1-\delta $$ Moreover $f_n(1)=0.$ But $$\int\limits_0^1f_n(x)\,dx =1-{n+1\over 2n+1}\to {1\over 2}$$