Show that the characteristic function of $\mathbb{Q}$ is Lebesgue integrable.

Question

Show that the characteristic function of $\mathbb{Q}$ is Lebesgue integrable.

I've shown that it isn't Reimann integrable, but I'm stuck on showing it is Lebesgue integrable. Any help would be much appreciated.

Answer 1

You have that $\mathbb{Q}$ is countable, so you can write $\mathbb{Q}=\bigcup_{n\geq 0}\{x_n\}.$ As a countable union of borelians sets, $\mathbb{Q}$ is a borelian set and so $\mathbb{1}_\mathbb{Q}$ is mesurable : by definition $$\int_\mathbb{R}\mathbb{1}_\mathbb{Q}\,d\mu=\mu(\mathbb{Q})=\sum_{n\geq 0}\mu(\{x_n\})=\sum_{n\geq0}0=0.$$

Answer 2

You have to check that the preimages of open semilines $(a,\infty)$ are measurable. As the function takes only two values, the only possible preimages are $\emptyset$, $\mathbb Q$, and $\mathbb R$.