Is this true that the sequences evenly sampled from functions in $L^p$ are in $l_p$?

Question

Is the following statement true ?

Given a function $f(x)$ in $L^p(\mathbb{R})$, $1\le p<\infty$, the sequence $\{f(x+kT)\}_{k\in\mathbb{Z}}$ is in $l_p$ for any $x$ and $T$ if every number in the sequence is finite.

Answer 1

This statement is not true for every $x\in\mathbb{R}$, however it is true for almost every $x\in\mathbb{R}$:

Suppose that there is a measurable $E\subset\mathbb{R}$ with $\mathrm{Leb}(E)>0$ such that $(f(x+kT))_{k}\notin\ell^p$ for all $x\in E$. W.l.o.g. assume that $E\cap (T+E)=\emptyset$. Then $$ \int_{\mathbb R}|f(x)|^p\,dx\geq \sum_{k\in\mathbb Z}\int_{kT+E}|f(x)|^p\,dx=\sum_{k\in\mathbb Z}\int_E|f(x-kT)|^p\,dx=\int_E\sum_{k\in\mathbb Z}|f(x-kT)|^p\,dx=\infty $$ contradiciting the assumption $f\in L^p$.