If $|f_n| \to 0$ and $f_n$ are integrable, is it true that $\int |f_n| \to 0$?

Question

If $|f_n| \to 0$ and $f_n$ are integrable, is it true that $\int |f_n| \to 0$?

Answer 1

Try $$ f_n=\frac1n\,1_{[0,n^2]}. $$

For an example on a bounded interval, one can take $$ f_n=n\,1_{[1/n,2/n]}. $$ Then $f_n\to0$, but $\int f_n=1$ for all $n$.