Construct a sequence of functions in $L^p(\mathbb{R})$ such that $f_n$ converges to zero in $L^p$.

Question

I'm trying to find a sequence of functions $f_n\in L^p(\mathbb{R})$, with $1\leq p < \infty$ such that $\lVert f_n \rVert_{L^p}\to 0$. Moreover for all $x\in \mathbb{R}$ the sequence satisfy two conditions $$\liminf f_n(x)= 0 \ \text{and}\ \limsup f_n(x)= \infty.$$

I think in use some sequence based on characteristic function, but i can't find the sequence.

Answer 1

I'll only give a solution for $L^1(\Bbb R)$. But just by suitable normalizing, this works for other $p$ as well.

Let's first focus on $[0,1]$. Consider,

$f_{1,0}:=\chi_{[0,1]}$

$f_{2,0}:=\sqrt{2} \chi_{[0,\frac{1}{2}]}$, $f_{3,0}:=\sqrt{2} \chi_{[\frac{1}{2},1]}$

$f_{4,0}:= \sqrt{3} \chi_{[0,\frac{1}{3}]}$, $f_{5,0}:= \sqrt{3} \chi_{[\frac{1}{3},\frac{2}{3}]}$, $f_{6,0}:= \sqrt{3} \chi_{[\frac{2}{3},1]}$

$\vdots$

So for a general $n>1$ we have a unique $m \in \Bbb N$ such that $\frac{m(m-1)}{2} <n \le \frac{m(m+1)}{2}$ and $f_{n,0}=\sqrt{m}\chi_{E_{n_i}}$ where $E_{n_i}$ is some subinterval of $[0,1]$ with $|E_{n_i}| = \frac{1}{m}$ and thus $$\int_{\Bbb R}|f_{n,0}(x)|dx =\int_{E_{n_i}}\sqrt{m}dx=\frac{1}{\sqrt{m}} \to 0 \text{ as } m \to \infty \text{ ( and so does } n \to \infty )$$

Note that given any point $x \in [0,1]$ it lies in infinitely many of the above sets and also does not lie in the rest of infinitely many of them!

Now there is nothing special about $[0,1]$, as $$\Bbb R=\cup_{k \in \Bbb Z}[k,k+1]$$ we can do the same business for each of these integral intervals. Then you consider this whole family of functions i.e. taking all $f_{n,k}$ from all $[k,k+1]$ and then for this family say $\{g_j\}_{j \in \Bbb N}$ you have,

Exercises

1. Check that you get the desired properties on $\liminf g_j $ and $\limsup g_j$.
2. Normalize the coefficients for $1<p<\infty$

Answer 2

Let $f_n = n^{c} I_{E_n}$ where $E_n$ is a measurable set of measure $ \le 1/n$ and $I_{E_n}$ its indicator function. We have $\|f_n\|_p^p = \le n^{pc-1} \to 0$ if $c < 1/p$. Now all you need to do is find a sequence $E_n$ such that every point of $\mathbb R$ is in infinitely many $E_n$. You can let $E_n = \cup_{k=-\infty}^\infty F_{k,n}$ where $F_{k,n}$ is an interval of length $2^{-|k|-2}/n$ in $[k,k+1]$. And for each $k$, let the intervals $F_{k,n}$ "walk" back and forth across $[k,k+1]$. Since $\sum_n 1/n$ diverges, you can do this so that each point is covered infinitely many times.