Limit of a Lebesgue integrable function

Question

I am having problems with following question:

Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$, and let $f: \mathbb{R} \to \mathbb{R}$ be Lebesgue integrable. Show that $\lim_{n \to \infty} f(x + n) = 0 = \lim_{n \to \infty} f(x - n)$ for $\lambda$-almost every $x \in \mathbb{R}$.

I tried to show that the set of all $x \in \mathbb{R}$ such that $\lim_{n \to \infty} f(x + n) \neq 0$ is a set with measure zero. But I have trouble showing that. I would be glad to get some pointers or ideas.

Answer 1

After switching the sum and integral and making the change of variables $y=x-k$ we get

$$ \int|f(x)|dx = \sum_{k\in\mathbb{Z}} \int_k^{k+1} |f(x)|dx = \int_0^1 \sum_{k\in\mathbb{Z}} |f(y+k)| dy$$

So the rightmost integral is finite. Since integrable functions are finite almost everywhere, we have that the sum on the right converges a.e., and hence that $|f(y+k)| \to 0$ as $|k|\to\infty$ for almost every $y\in[0,1]$ and hence almost every $y\in\mathbb{R}$. In fact, this convergence to zero happens at a summable rate.

Answer 2

Let's assume we'd have $S\subseteq \mathbb R $ with $\lambda(S)>0$ such that $$\forall x\in S:\quad \lim_{n \to \infty} f(x + n) = 0\lor 0 = \lim_{n \to \infty} f(x - n)$$

Then at least one of the sets $\{x\in\mathbb R\mid\lim_{n \to \infty} f(x + n) = 0\}$ and $\{x\in\mathbb R\mid\lim_{n \to \infty} f(x - n)=0\}$ must have positive measure.

Assume wlog that $\{x\in\mathbb R\mid\lim_{n \to \infty} f(x + n) = 0\}$ has positive measure.

This means that for each $x\in S$ we have a constant $K_x>0$ such that

$$\exists N>0:\forall n\in\mathbb N:\quad n>N\implies f(x+n)\ge K_x\tag{1}$$

With this we have: $$ \int_\mathbb R |f(x)| d\lambda \ge\int_{\{s+n\mid s\in S, n\in\mathbb N\}} |f(x)| d\lambda = \sum_{n\in \mathbb N} \int_{S+n} |f(x)| d\lambda $$

It now suffices to show that $\lim_{n\to\infty}\int_{S+n} |f(x)| d\lambda >0$.

To use Fatou's Lemma, we rewrite it as $$\lim_{n\to\infty}\int_{S+n} |f(x)| = \lim_{n\to\infty}\int_{S} |f_n|$$ with $f_n:S\to\mathbb R,\quad x\mapsto x+n$.

Using Fatou's Lemma: $$ \lim_{n\to\infty}\int_{S} |f_n| = \lim\inf_{n\to\infty}\int_{S} |f_n| \ge \int_{S} \lim\inf_{n\to\infty}|f_n| $$

Per definition of $f_n$ and $(1)$ we have that $\lim\inf_{n\to\infty}|f_n|>0$, and thus $\int_{S} \lim\inf_{n\to\infty}|f_n|>0$.