$f_{n} \overset{\left\lVert .\right\rVert{.}_{1}}{\longmapsto} 0$ in $L^{1}([0,1])$ but not converging to $0$ almost everywhere.

Question

I'd like to understand the costruction of the following function we should statisfy the requests of the title. We define $S_{n} := \sum\limits_{k=1}^{n}\frac{1}{k}$, with $a_{n} := S_{n} - \lfloor S_{n}\rfloor = S_{n} \hspace{0.1cm} \mbox{mod}\hspace{0.1cm}(1)$ and we set $f_{n} := \chi_{[a_{n},a_{n+1}]}$.

It holds that $|a_{n}-a_{n+1}| = \frac{1}{n}$ and that $f_{n} \overset{\left\lVert .\right\rVert{.}_{1}}{\longmapsto} 0$ in $L^{1}([0,1])$, but not converging to $0$ almost everywhere, it should be true that $f_{n} = 1$ for infinite values.

What I don't understand is the costruction of $S_{n}$ which I think carries all the other properties I should verify. In particular it's not obvious to me that $|a_{n}-a_{n+1}| = \frac{1}{n}$ and $f_{n} = 1$ for infinite values.

Any help or hint would be appreciated.

https://math.stackexchange.com/questions/1281421/convergence-in-lp-implies-convergence-in-measure — Sean Nemetz, Jun 18 '20 at 16:22
@SeanNemetz $\forall \hspace{0.1cm} n_{0} \hspace{0.1cm} \exists \hspace{0.1cm} n \geq n_{0} : f_{n} = 1$ — jacopoburelli, Jun 18 '20 at 16:37

score 1 · Accepted Answer · answered Jun 18 '20 at 20:58

1

If what you want is an example where $f_n\to 0$ in $L^1$ but $f_n$ does not converge pointwise to $0$ a.e., then consider the "typewriter" sequence:

$$\chi_{[0,1]}, \chi_{[0,1/2]}, \chi_{[1/2,1]}, \chi_{[0,1/3]},\chi_{[1/3,2/3]}, \chi_{[2/3,1]}, \dots $$

answered Jun 18 '20 at 20:58

zhw.

107,943

I think they are the same sequence but I don't know how to prove it – jacopoburelli Jun 18 '20 at 21:25

$f_{n} \overset{\left\lVert .\right\rVert{.}_{1}}{\longmapsto} 0$ in $L^{1}([0,1])$ but not converging to $0$ almost everywhere.

1 Answers1