Suppose $f$ is measurable. For each natural number $n$, define $ X_n = \{x \in X : f(x) \geq 1/n\}.$ Why is $\bigcup_{n=1}^\infty X_n = \{x\in X : f(x) \gt 0\}?$ Suppose $f$ is measurable and integrable. How can one write the measurable set $\{x: f(x) \lt 0\}$ as a countable union of measurable sets?
Can these be generalized?
Thanks.