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.
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