There is a theorem in my textbook (not on English). The theorem states the following:
We have a sequence of measurable functions $f_k$. They are finite almost everywhere on the set $E\subset X$ where $\mu E< +\infty$. $f_k\rightarrow 0$ almost everywhere on $E$. Then there is exists an increasing sequence of positive numbesrs $\lambda_k\rightarrow +\infty$ such that $\lambda_k f_k(x)\rightarrow 0$ almost everywhere on $E$.
Then there is written that this property is called ..."stability of convergence"...or something like that. I can not search this term in Google.
Question: can someone say what is this property on English? Or can some one promt where I can find it?
Since $\mu(E)<\infty$, $g_n\to 0$ a.e. on $E$ iff $\lim_{n\to\infty}\mu(\{E:\sup_{m\ge n}|g_m|>\epsilon\})= 0$ for any $\epsilon>0$. Now, construct a sequence $\{\lambda_n\}$ recursively. Set $\lambda_n=k^{-1}$ for $N_k\le n< N_{k+1}$, where $N_k$ is s.t. $$ \mu\left(\left\{E:\sup_{n\ge N_k}|f_n|>k^{-1}\right\}\right)<k^{-1}. $$