Let $E\subset\mathbb{R}$ be measurable and for each $n\in\mathbb{N}$, let $f_n:E\rightarrow\mathbb{R}$ be measurable. For each $\varepsilon > 0$, define $E_n(\varepsilon) = \{x\in E : |f_n(x)|\geq \varepsilon\}$. Show tht if $\sum_{n=1}^{\infty}m(E_n(\varepsilon))<\infty$ for every $\varepsilon >0$, then $\lim_{n\rightarrow\infty}f_n(x)=0$ for almost every $x\in E$.
So I've been thinking about this problem for a while now and I just don't see how it even makes sense. What should my set of measure zero be that I'm removing from $E$?
By going far enough down the sequence $\{E_n(\varepsilon)\}$, and then taking the union of the remaining sets in its tail: $\cup_{n=k}^{\infty}E_n$, I can make the measure of this union as small as I like, but ultimately it is fixed, and must be removed from $E$ prior to taking the $\lim_{n\rightarrow\infty}f_n(x)$, and thus it will never necessarily have measure zero.
The only reasonable sets of measure zero I see would be some infinite intersection of sets in the sequence $\{E_n(\varepsilon)\}$. But this seems to make my set of measure zero too restrictive, and I see no way of doing this and still avoiding having points in my remaining set which map under $f_n$, for an infinite number of $n$, to values whose magnitude is greater than $\varepsilon$.
Am I missing something? Thanks.