Let $f$ be Lebesgue integrable on $\mathbb{R}$. Show that $ \sum_{n=1}^{\infty} f(x+n) $ converges almost everywhere.

Question

Let $f$ be Lebesgue integrable on $\mathbb{R}$. Show that $ \sum_{n=1}^{\infty} f(x+n) $ converges almost everywhere.

I was thinking that maybe the finite sum could be compared to a simple function and maybe this could lead somewhere, but i've doubt this since I don't know what the level sets look like for the $f(i)'s$. The result seems a bit obvious but I don't have a good idea.

Any helpful hints or help would be great

Answer 1

We can assume without loss of generality that $f$ is non-negative. Define $S_N(x):=\sum_{j=1}^nf(x+j)$. We can show that $$\int_{[0,1]}S_N(x)\mathrm d\lambda(x)= \int_1^{N+1}f(x)d\lambda(x)\leqslant \int_{\mathbb R}f(x)d\lambda(x),$$ hence by Fatou's lemma, $S(x):=\sum_{n=1}^\infty f(x+n)$ is integrable on $[0,1]$. This proves that there exists $N_0\subset [0,1]$ of null measure such that $S(x)\in\mathbb R$ if $x\in [0,1]\setminus N_0$.

Now, we can argue similarly in order to prove that $S$ is integrable over $[k,k+1]$. This will produce an $N_k$.