Calculate the Lebesgue integral Calculate the Lebesgue integral 
$$ \int_{[0,\pi]}^{} \sin x\: \epsilon[\mathbb{R}\setminus\mathbb{Q}] d\mu,$$
where is $\epsilon $ is the characteristic function.
I tried this way $ \int_{\mathsf ([0,\pi ] \cap I) }^{} \sin x \: d\mu$...
$\mathsf ([0,\pi ] \cap I)$ = $[0,\pi ]$.. 
$ \int^{\pi}_0 \sin x \: d\mu=-\cos(\pi)+\cos(0)=2$
Is this correct?

You are not far, just use that $m([0,\pi])=m((R-Q) \cap [0,\pi])$
And the fact that if $m(E)=0$ then $\int_Ef=0$ for some measurable $E$ and $f$ integrable.(where $f$ can also be negative)

$\int_{\mathbb{Q \cap [0,\pi]}}\sin{x}=0$ because $m( \mathbb{Q} \cap[0,\pi])=0$
So  $$\int_{(\mathbb{R}-\mathbb{Q}) \cap [0,\pi]} \sin{x}=\int_0^{\pi}\sin{x}-\int_{\mathbb{Q \cap [0,\pi]}}\sin{x}=\int_0^{\pi}\sin{x}=2$$

