# Decreasing sequence of Lebesgue measurable functions that converge $0$ and that don't converge in mean

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.

• Consider $\frac f n$ where $f$ is any nonnegative measurable function which is not integrable. Oct 3 '19 at 9:19

Take $$f_n=1_{[n,+\infty)}$$
$$f_n$$ is deacreasing also