Let $f\in L^1(\mathbb{R})$, or in another word $\int_{\mathbb{R}}\rvert f(x)\rvert dx < \infty$. Show $$\int_{\mathbb{R}}\frac{1}{2h}\int_{x-h}^{x+h} |f(t)| \,dt\,dx < \infty$$ This inequality gives us the condition to apply Fubini's theorem to the double integral. But I am having difficulties to show the above inequality.
The first integral converges to $f(x)$ if $h\rightarrow 0$, but this perspective does not provide enough information to say about the integrability for the double integral.
Thanks for any help or advice!