I've come across a problem which states:
Given a sequence of integrable functions $\{f_k\}$ ($k≥1$) on $[0,1]$ with the property that $||f_k||_1 ≤ \frac{1}{2^k} $, then $f_k \rightarrow 0$ pointwise almost everywhere on $[0,1]$.
I'm not exactly sure how to proceed. I attempted to show this via contradiction, but couldn't see how to finish the argument.
Best Answer
For a sequence of non negative measurable functions we can interchange infinite sum with integral, it follows from monotone convergence. So we have:
$\int_0^1(\sum_{k=1}^\infty |f_k|) dx=\sum_{k=1}^\infty\int_0^1 |f_k|dx=\sum_{k=1}^\infty ||f_k||_1<\infty$
So the integral of the non negative function $f:=\sum_{k=1}^\infty |f_k|$ is finite, and hence $f$ has to be finite almost everywhere. But at every point where the series is convergent we have $f_k\to 0$. So it happens almost everywhere.