Convergence in $L^1$ norm, but not point-wise a.e.

Question

As part of a course assignment, I'm asked to find a sequence of functions that converges in $L^1(\Omega \subset \mathbb{R})$, yet does not converge point-wise a.e. My thought is that such a sequence would have to be some sort of sliding function that moves along, say, $[0,1]$, so that each point has a constant height infinitely often. I've tried to illustrate what I mean below:

Iteratively,

Though, I'm not quite certain if this is the right idea. I can't seem to think of anything else that would behave in this manner, converging to $0$ in integral, but not point-wise. My issue currently, if this is the correct idea, is that I have absolutely no idea how to explicitly write such functions down.

The one's I have depicted would be some sort of shifting of the bump function involving $e^x$, yet I don't know what shifts would be correct, along with what scaling.

Answer 1

Your idea is perfect; a compact way of writing something that behaves as Bungo's sequence is the following: $$f_n(x) = \chi_{[0,1]}(2^mx - j),\ n = 2^m + j,\ 0 \le j \le 2^m - 1.$$

Here the $m$ is shrinking the length of the interval in which the function is $1$ and the $j$ is responsible for translating this interval.

Answer 2

After a little work, I was able to write Bungo's answer explicitly as the sequence of functions $$f_n(x) = \chi_{[\frac{n-2^m}{2^m},\frac{n-2^m+1}{2^m}]}$$ with $m \geq 0$ and $2^m \leq n < 2^{m+1}$.

I'm adding it as an answer for anyone who searches this question in the future, as this may be an easier interpretation for some.