Let $(X,\mu,\mathcal{A})$ be a finite measure space and $f_n$ measurable functions such that $f_n \to 0 $ almost everywhere.
Show that exists a sequence $a_n \to +\infty$ auch that $a_nf_n \to 0$ a.e.
I managed (by using the Borel-Cantelli lemma) to find a subsequence $a_{n_m}$ such that $a_{n_m}f_{n_m} \to 0 $ a.e using the convergence in measure(since we have convergence a.e),but i could not solve it.
Can someone give me a hint?
I do not seek a full solution.
Thank in advance.
Best Answer
Assuming that $\mu(X)<\infty$, $f_n\to 0$ a.e. iff for every $\epsilon>0$, $\mu(\sup_{k\ge n}|f_k|>\epsilon)\to 0$ as $n\to\infty$. In this case $\{a_n\}$ can be constructed as follows. Pick the sequence $n_j$ s.t. $\mu(\sup_{k\ge n_j}|f_k|>\epsilon/j)\le j^{-1}$ and for each $j\ge 1$, set $a_{n_j}=\cdots=a_{n_{j+1}-1}=j$.