Example of sequence of integrable maps such that integral smaller than $1/n$ but doesn't converge to zero almost everywhere

Question

I am looking for a measure space $(\Omega,\mathcal{M},\mu)$ and a sequence of integrable functions $(f_n)$ in $\mathcal{L}^1(\Omega,\mathcal{M},\mu)$ with the property that $\int_\Omega |f_n|d\mu \leq \frac{1}{n}$ but such that the following statement is false:

• $f_n(x) \to 0$ for almost all $x\in \Omega$.

I've been having a lot of trouble coming up with a concrete counterexample.

Thanks in advance!

Answer 1

If you know some probability theory you can take an i.i.d. sequence of random variables $X_n$ which are uniformly distributed over $[0,1]$ and apply the Borel-Cantelli lemma to the indicator functions $f_n$ of $A_n=\{ X_n<1/n\}$ so that $\sum\limits_{n=1}^\infty P(A_n)=\infty$. Borel-Cantelli yields $P(\lim\sup A_n)=1$, i.e., $P$-almost every point of the underlying probability space belongs to infinitely many $A_n$ so that $f_n$ does not converge to $0$ almost surely.

The idea of the typewriter sequence mentioned by Andrew in the comments is of course similar.