I am trying to understand and prove the following:

Let $\{f_n\}$ be a sequence of measurable functions on $[0,1]$ that converges to $0$ a.e. on $[0,1]$.

Then prove there exists a sequence of positive numbers $\{A_n\}$ that tends to infinity where the sequence $\{A_nf_n\}$ converges to $0$ a.e. on $[0,1]$.

I am new to this group and any help or hints for this proof would be greatly appreciated. I believe that using Littlewood's principles is useful in this proof, but I am lost on where to begin.


I don't remember Littlewood's principles, but Egorov's theorem comes to mind. For every $k$ there is a set $E_k$ of measure at most $2^{-k}$ outside of which the sequence converges to $0$ uniformly. Therefore, there is $n_k$ such that $|f_n \chi_{E_k^c}|\le 4^{-k}$ for $n\ge n_k$. The sequence $\{n_k\}$ can be arranged to be strictly increasing. Let $A_n=2^k$ for $n_k\le n<n_{k+1}$. Then $$\sup |A_n f_n \chi_{E_k^c}|\le 2^{-k},\quad n_k\le n<n_{k+1} \tag{1}$$ Finally, use the fact that $\sum |E_k|<\infty $ to argue that a.e. point $x\in [0,1]$ is only in finitely many $E_k$, hence $A_n f_n(x)\to 0$. (This is known as Borel–Cantelli lemma, but easy to prove from basic principles if you don't know it.)


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