# 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 $$f$$.

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 Here the change of order of summation and integration can be justified by either monotone convegernce, or by Fubini's theorem.

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

Answered by Oliver Diaz on January 7, 2022

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