Let $p\in[1,\infty)$ and $(f_n)_{n\in\mathbb{N}}\subset L^{p}(\mathbb{R})$ such that $\|f_n\|_{p}\leq n^{-2}$ for all $n\in\mathbb{N}.$ Does $(f_n)_{n\in\mathbb{N}}$ necessarily converge pointwise a.e.? (Proof or counterexample.)
$\textbf{Attempt:}$ I think one can construct a counterexample to this. I was considering the following construction done by Saz in this post: https://math.stackexchange.com/a/3132844/595519.
I provide the details below:
Consider the following sequence of intervals $$[-1,0], [0,1],\left[-2,-\tfrac{3}{2}\right], \left[-\tfrac{3}{2},-1\right], \left[-1,-\tfrac{1}{2}\right],\left[-\tfrac{1}{2},0\right],\left[0,\tfrac{1}{2}\right],\left[\tfrac{1}{2},1\right],\left[1,\tfrac{3}{2}\right],\left[\tfrac{3}{2},2\right],\left[-3,-\tfrac{8}{3}\right],\left[-\tfrac{8}{3},-\tfrac{7}{3}\right],\dots,\left[\tfrac{7}{3},\tfrac{8}{3}\right],\left[\tfrac{8}{3},3\right],\dots$$ Denote the $n$th interval in the sequence by $I_n.$ By construction, it follows that $m(I_n)\rightarrow 0$ as $n\rightarrow\infty.$ Now denote the characteristic function of the $n$th interval above by $1_{I_n},$ and put $f_n(x)=n^{-3}\cdot m(I_n)^{-1/p}\cdot1_{I_n}(x)$. It follows that $$\left(\int_{\mathbb{R}}|f_n(x)|^p\,dx\right)^{1/p}=n^{-3},$$ and it follows that $\|f_n-0\|_{p}\rightarrow 0$ as $n\rightarrow\infty,$ so $f_n\rightarrow 0$ in the $L^p$-norm.
However, I note that the sequence $(f_n)_{n\in\mathbb{N}}$ I have constructed above fails, since it actually does converge pointwise to $f\equiv 0,$ and even worse, it does so everywhere.
Is there any way to save this example, or should I scratch it and try something else? Finally, is there a way to use decaying exponentials in this problem to make the sequence decrease fast enough?
Thank you for time and appreciate any feedback.
$\sum _n\int |f_n|^{p} \leq \sum _n \frac 1 {n^{2p}} <\infty$. Hence $\sum _n |f_n|^{p} <\infty$ almost eveywhere and $f_n \to 0$ almost eveywhere.