Example of a Lebesgue integrable function under certain conditions

Question

I need to find a sequence $(f_k)$ of Lebesgue integrable functions such that $f_k \to 0$ almost everywhere but $\lim _{k\to \infty} \int |f_k| \ne 0$.

Here is an example that I thought: $f_k(x) = g_{[ 0, k)} (1/kx) $ . Here $g_{[ 0, k)}$ is a characteristic function . Is it right ?

Answer 1

Both $$f_k = \frac{1}{k}g_{[0,k]}$$ and $$f_k =k g_{[0,\frac{1}{k}]}$$ work as examples. The integrals of both are $1$, yet one goes up to "vertical infinity" while the other flattens to "horizontal infinity"