Are $(\alpha ,\beta]$ and $[\alpha ,\beta)$ lebesgue measurable?

Question

It is true that all open sets are lebesgue measurable, but what about not open/not close intervals?

Answer 1

$$(\alpha ,\beta ]=\bigcap_{n\geq 1}\left(\alpha ,\beta +\frac{1}{n}\right).$$

So, as you can see, it's even better than just Lebesgue measurable.

Answer 2

Yes. Every singleton is a complement of an open set, hence measurable. Therefore $(\alpha,\beta] = (\alpha,\beta)\cup\{\beta\}$ is a union of two measurable sets.