10:25 PM
Suppose $f,g : [-\infty, \infty] \to [- \infty, \infty]$ are measurable. Prove that the sets $\{x \mid f(x) < g(x) \}$ and $\{x \mid f(x) = g(x)\}$ are measurable.
The first is measurable because $\{x \mid f(x) < g(x) \} = \bigcup_{q \in \Bbb{Q}} \bigg[ \{x \mid f(x) < q \} \cap \{x \mid q < g(x)\} \bigg]$; i.e., the set is the countable union of measurable sets.
The second set is measurable because $\{x \mid f(x) = g(x)\} = \bigg[ f^{-1}(-\infty) \cap g^{-1}(-\infty) \bigg] \cup \bigg[f^{-1}(\infty) \cap g^{-1}(\infty) \bigg] \cup (f-g)^{-1}(0)$, which means the set is the union of measurable sets and $f-g$ is a measurable function.
