Consider the function
$$ f(x) = \frac{1}{\sin(x)} $$ a) Is $f$ Lebesgue measurable? b) is $f$ integrable on $\mathbb{R}$?
for a) I have if $A=\{y\in\mathbb{R} : \frac{1}{\sin(y)} = 0 \} \cup \{0\}$ is of measure 0 so the function is measurable.
for b) I am trying to solve using approximation lemma but it is not going well.