I have following set $$M := \left\{ (x,y) \in \mathbb{R}^2 \mid \exists q \in \mathbb{Q}: \frac{q}{\pi} \leq \sin(xy)<\cos(y)+q^4\right\}$$
and I am supposed to verify that the set $M$ is Lebesgue measurable but how do I calculate the Lebesgue measure of an explicit given set?
In fact, $M$ is an open set.
For all $q \in \mathbb{Q}$ define $A_q = \{ (x,y) \in \mathbb{R}^2 : q < \pi \sin (xy)\}$, $B_q= \{ (x,y) \in \mathbb{R}^2 : \sin (xy) - \cos y < q^4\} $. Since $\pi \sin(xy)$ and $\sin(xy) - \cos y$ are continuous functions, $A_q$ and $B_q$ are open sets. Finally
$$M = \bigcup_{q \in \mathbb{Q}} A_q \cap B_q$$
and this is a countable union of open sets, so it is open, hence measurable.
However, in order to show that $M$ is measurable, you need just that $\pi \sin(xy)$ and $\sin(xy) - \cos y$ are measurable functions, so that $A_q$ and $B_q$ are measurable sets.