example of a sequence $f_n$, $n=1,2...$ of integrable functions converging to $f$ s.t limit of integral of $f_n$ does not exist

Question

Is there an example of a sequence of of functions $f_n$ converging to a function $f$ such that $f_n$, $n=1,2...$ are integrable and nonegative and their integral over a measurable set $A$ is less than a finite $K$ for all $n$ , however the limit of integral of $f_n$ on $A$ does not exist?

I am just trying to understand why Fatou theorem has $\liminf$ of integral of $f_n$.

Thank you

Answer 1

$f_n = \frac{1}{n}\cdot \chi_{[0,n]}$ for $n$ even and $f_n \equiv 0$ for $n$ odd.

Then $f_n \to f$ pointwise, but $\int f_n = 1$ for $n$ even and $\int f_n = 0$ for $n$ odd (everything with the usual Lebesgue measure).