I am trying to prove the following proposition:
Let $f \in L^p(\mathbb{R})$, then for almost every $x \in [0,1]$, we have:
$$\lim_{n \to \infty} f(x+n) = 0$$
Note that $|f(x)|^p = \sum_{-\infty}^\infty \chi_{[n,n+1)} |f_n(x)|^p$, thus, we can apply Tonelli and a change of variables to get that
$$\int |f(x)|^p dx= \sum_{-\infty}^\infty \int \chi_{[n,n+1)} |f_n(x)|^p dx = \sum_{-\infty}^\infty \int_{n}^{n+1} |f_n|^p dx = \sum_{-\infty}^\infty \int_{0}^{1}|f_n(x+n)|^p dx$$
Thus, since $f \in L^p(\mathbb{R})$, the limit of the integrals $\int_{0}^{1}|f_n(x+n)|^p dx$ must converge to $0$. From this, we can use Fatou's Lemma to conclude that, for almost every $x\in[0,1]$, we have
$$ \liminf_{n \to \infty} f(x+n) = 0.$$
Thus, if the limit we are looking for exists, then the result must follow. However, I am not entirely sure that it must exist. On the other hand, I have tried to come up with counterexamples but couldn't.
Any thoughts or counterexamples are welcome. Thanks!
For $\varepsilon>0$ define $$E_{n,\varepsilon} =\{ x\in [0,1) \ : \ \vert f(x+n)\vert \geq \varepsilon\}.$$
Then we have $$ \Vert f \Vert_{L^p(\mathbb{R})}^p\geq \sum_{n\in \mathbb{N}} \int_0^1 \vert f(x+n)\vert^p dx \geq \sum_{n\in \mathbb{N}} \varepsilon^p \lambda(E_{n,\varepsilon})),$$ where $\lambda$ denotes the Lebesgue measure. As $f\in L^p(\mathbb{R})$, we get that $$\sum_{n\in \mathbb{N}} \lambda(E_{n,\varepsilon})<\infty.$$ By the Borel-Cantelli lemma, we get that $$\lambda(\limsup_{n\rightarrow\infty} E_{n,\varepsilon})=0,$$ which translates to $$\limsup_{n\rightarrow \infty} \vert f(x+n)\vert \leq \varepsilon$$ almost surely. Now take the union over countably many $\varepsilon_m\rightarrow 0^+$ to conclude that $\limsup_{n\rightarrow\infty} \vert f(x+n)\vert =0$ almost surely. For example we can do it this way \begin{align*} &\lambda(\{x\in [0,1) \ :\ \limsup_{n\rightarrow\infty} \vert f(x+n)\vert >0\}) \\ &\leq \sum_{m\in \mathbb{N}}\lambda(\{x\in [0,1) \ : \ \limsup_{n\rightarrow \infty} \vert f(x+n)\vert \geq 1/(2m)\}) \\ &\leq\sum_{m\in \mathbb{N}} \lambda (\limsup_{n\rightarrow \infty} E_{n, 1/m})=0. \end{align*}
Of course this only works for $1\leq p<\infty$. For $p=\infty$ this miserably fails, $$f(x)=\sum_{n\in \mathbb{N}} \chi_{[2n, 2n+1]}(x)$$ is an example in $L^\infty(\mathbb{R})$ where $\lim_{n\rightarrow \infty} f(x+n)$ does not exist for a single $x\in [0,1]$. In fact, in the example above, we have $\limsup_{n\rightarrow \infty} f(x+n)=1$ for all $x\in \mathbb{R}$.