Does there exist a decreasing sequence of Lebesgue-measurable non-negative functions $(f_n)_{n}$ such that $f_n \to 0$ pointwise on $\mathbb{R}$, but $f_n$ does not converge to $0$ in mean, by which I mean $$ \int_\mathbb{R} f_n d\mathcal{L}^1 \not\to 0 \text{ as } n \to \infty ?$$
I tried finding a counterexample, but I did not succeed. There is a theorem that says that if $|f_n| \leq g$ for every $n$ and $\displaystyle \int_\mathbb{R} |g| d\mathcal{L}^1 < \infty,$ then $\displaystyle \int_\mathbb{R} f_nd\mathcal{L}^1 \to 0$ as $n \to \infty$, but here we don't have the conditions that the sequence of functions is dominated by an integrable function.
Best Answer
Take $f_n=1_{[n,+\infty)}$
$f_n$ is deacreasing also