Prove the series converges almost everywhere

Question

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 $f$.

Oliver Diaz · 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.

Prove the series converges almost everywhere

One Answer

Add your own answers!

Ask a Question