If $\{O_j\}_{j=1}^\infty$ are open intervals in $\mathbb R$ and $\mathscr O = \bigcup \limits_{j=1}^{\infty} O_j$, is it true that
$$\int_{\mathbb R} \chi_{\mathscr O} dx \leq \sum_{j=1}^{\infty} \int_{a_i}^{b_i} \chi_{\mathscr{O}} \, dx?$$
These integrals are Riemann integrals. It is obvious that this is true for lebesgue integral, but I can't see how to show it for Riemann Integrals. (See comments for discussion)