# sufficient condition for mean convergence in $L^1$:prove or disprove

Let $$f_n \; \colon (0,1) \to \mathbb{R}$$ a sequence of Lebesgue integrable functions which converges almost everywhere in $$(0,1)$$ to $$0$$. Prove or Disprove: if there exists $$p \in (1,+ \infty)$$ such that $$(f_n)$$ is bounded in $$L^p(0,1)$$,then $$\lim_{ n \to \infty} \int_0^1|f_n(x)| \, dx =0$$.

I think the property is true and Egorov's theorem will help here. Fix $$\varepsilon >0$$. Since $$f_n \to 0$$ a.e., there is a set $$E \subset (0,1)$$ such that $$\lvert E\rvert < \varepsilon$$ and $$f_n \to 0$$ uniformly on $$(0,1)\setminus E$$. Now fix $$\delta > 0$$ and $$N$$ large enough that $$\lvert f_n \rvert < \delta$$ on $$(0,1)\setminus E$$ whenever $$n \ge N$$. Then for $$n \ge N$$, \begin{align*} \int^1_0 \lvert f_n(x)\rvert dx &= \int_{(0,1)\setminus E} \lvert f_n(x)\rvert dx + \int_E \lvert f_n(x)\rvert dx \\ &\le \delta \int_{(0,1)\setminus E}dx + \lvert E \rvert^{1/q}\|f_n\|_{L^p(0,1)}\\ &\le \delta + \varepsilon^{1/q}M \end{align*} where $$M$$ is the bound on $$f_n$$ in $$L^p(0,1)$$, and $$q$$ is the dual exponent to $$p$$. Since $$\varepsilon$$ and $$\delta$$ are arbitrary, this shows that $$\int^1_0 \lvert f_n(x)\rvert dx \to 0.$$

For future readers: this answer is a failed attempt at a counterargument. I’m leaving it here instead of deleting it since I think my faults here can be instructive. Read the comments below for more detail.

Here’s a modification to what I posted earlier: let $$f_n:(0,1)\to\mathbb{R}$$ be defined by

$$f_n(x) =\begin{cases} n\ln(n), & \text{if} \;\; x\in\left(0,\frac{1}{n}\right)\\ 0, &\text{otherwise} \end{cases}.$$

We see that the $$f_n\to0$$ point wise, however we find that

$$\int_0^1 |f_n|dx = \frac{1}{n}\cdot n\ln(n) = \ln(n).$$

• hmmm, interesting. I posted what I think is a valid proof, so either this sequence isn't bounded in $L^p$ for any $p > 1$, or I made a mistake. EDIT: This function sequence is not bounded in $L^p$ since $\|f_n\|^p_{L^p} = n^{p-1}(\ln(n))^p$ Apr 11, 2020 at 22:17
• @User8128 I believe these functions are $L^p$ bounded because they are essentially step functions. I believe we can show that $||f_n||_p= n^{\frac{p-1}{p}}\ln(n)$. Apr 11, 2020 at 22:21
• @Mnifldz Each function is in $L^p$, but the sequence is not bounded in $L^p$ since $\|f_n\|_{L^p} \to \infty$ Apr 11, 2020 at 22:21
• @GiuseppeTenaglia The question seems to be this: assume that $f \to 0$ a.e and $(f_n)$ is bounded in $L^p$ for a fixed $p \in (1,\infty)$. Prove or disprove that $\|f\|_{L^1} \to 0$. If that is indeed the question, then my above answer proves that the property is true. Apr 11, 2020 at 22:26
• @Mnifldz no problem! Glad we could resolve this Apr 11, 2020 at 22:26