Let $\{f_k\}$ be measurable functions such that $\sum_{k=1}^\infty\int |f_k|d\mu<\infty$ Show that $\lim_{k\rightarrow 0}f_k(x)=0$ almost everywhere.

Question

Let $\{f_k\}$ be a sequence of measurable functions such that $$\sum_{k=1}^\infty\int |f_k|d\mu<\infty$$ Show that $\displaystyle\lim_{k\rightarrow 0}f_k(x)=0$ almos everywhere.

Since $\displaystyle\sum_{k=1}^\infty\int |f_k|d\mu<\infty$ I know that $\int |f_k|d\mu\rightarrow 0$, which means that the functions $f_k$ must be tending to zero almost everywhere, but I can't find a way to prove this.

user284331 · Accepted Answer · 2019-12-08T02:11:02.337

0

So by Tonelli, pairing with counting measure and $\mu$, one has \begin{align*} \int\sum|f_{k}|<\infty, \end{align*} and hence \begin{align*} \sum|f_{k}(x)|<\infty,~~~~\text{a.e.}~x, \end{align*} can you finish from here?

edited Dec 08 '19 at 02:11

answered Dec 08 '19 at 01:53

user284331

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Interchanging the sum with the integral is justified by monotone convergence, I presume? – Math1000 Dec 08 '19 at 01:59
When everything is positive, the sum and integral can be interchanged. See Tonelli's Theorem. – robjohn Dec 08 '19 at 02:00
let me try, I didn't thought of the monotone convergence theorem – AlephZero Dec 08 '19 at 02:02
I suppose Tonelli works as well. You just might want to make a note of that in your answer. – Math1000 Dec 08 '19 at 02:03

Let $\{f_k\}$ be measurable functions such that $\sum_{k=1}^\infty\int |f_k|d\mu<\infty$ Show that $\lim_{k\rightarrow 0}f_k(x)=0$ almost everywhere.

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