Sequence of Lebesgue integrable functions bounded in norm converges pointwise

Question

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.

Mark · Answer

For a sequence of non negative measurable functions we can interchange infinite sum with integral, it follows from monotone convergence. So we have:
$int_0^1(sum_{k=1}^infty |f_k|) dx=sum_{k=1}^inftyint_0^1 |f_k|dx=sum_{k=1}^infty ||f_k||_1<infty$
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 $f_kto 0$. So it happens almost everywhere.

Sequence of Lebesgue integrable functions bounded in norm converges pointwise

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