Let $f(x) = \sin(x)/x$. Consider the following:

$$\int\_{\mathbb{R}} f(x)dx = 
\lim\_{n\to\infty} \int\_{\mathbb{R}} f(x)\chi\_{[-n,n]}dx$$

so that each $\int\_{\mathbb{R}} f(x)\chi\_{[-n,n]}dx$ is Riemann integrable
and so it is Lebesgue integrable (integrating over compact space). Since
$\int\_{\mathbb{R}} f(x)\chi\_{[-n,n]}dx \to \int\_{\mathbb{R}} f(x)dx$ and
so by completeness of $L^1(\mathbb{R})$, we have that $f\in L^1(\mathbb{R})$ so in particular,
$\int\_{\mathbb{R}} f(x)dx < \infty$ in the Lebesgue integral sense.

I know I have made a mistake here somewhere but I am having trouble seeing *exactly* where. Any help is appreciated.