Prove the series converges almost everywhere

Mathematics Asked by Christopher Rose on January 7, 2022

Question: Given Lebesgue integrable $f: mathbb{R}rightarrow [0,infty)$, prove the following series converges almost everywhere on $mathbb{R}$:
$$varphi(x) = lim_{krightarrow infty} sum_{t=-k}^k f(t+x)$$

Attempt: Towards a contradiction suppose there is a non-null set $A$ such that for all $x in A$ we have $varphi(x)=infty$. Somehow I want to conclude that $int_A f=infty$ and contradict the integrability of $f$.

One Answer

Let $I=(-1/2,1/2]$. Then

$$int_{I}Big|sum_{kinmathbb{Z}}f(x+k),dxBig|leq int_{I}sum_{zinmathbb{Z}}|f(x+k)|,dx=sum_{kinmathbb{Z}}int_{I+k}|f(x)|,dx=|f|_1<infty$$ Here the change of order of summation and integration can be justified by either monotone convegernce, or by Fubini's theorem.

Thus $g(x)=sum_{kinmathbb{Z}}f(x+k)<infty$ a.s for all $xin I$, which can then be extended as a periodic function.

Answered by Oliver Diaz on January 7, 2022

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