Measurable functions and rational rays.

Question

I am reviewing for an exam, and the one problem I have not been able to make any headway on is this one:

Show that a function from a measure space $(X,\mathscr{M})$is measurable if and only if $\{x\in X:f(x)>q\}\in\mathscr{M}$ for all $q\in \mathbb{Q}$.

One direction is real easy, obviously. For the other, I suppose the idea is that we need to show that $E_x\in \mathscr{M}$ for $x\in\mathbb{R}\setminus \mathbb{Q}$, but I haven't been able to come up with any way of doing this, since there is not necessarily an ordering on $X$.

Answer 1

A function $f:X \to \mathbb{R}$ is Borel measurable iff $f^{-1}((a,\infty)) \in \mathcal{M}$ for all $a \in \mathbb{R}$. This is because the collection $\{ (a,\infty): a \in \mathbb{R} \}$ generates the $\sigma$-algebra of Borel sets on $\mathbb{R}$.

Now here is your hint: $$f^{-1}((a,\infty))=\bigcup_{q \in \mathbb{Q} \cap (a,\infty)} \{ x \in X : f(x)>q \}$$ This last union is countable. Finish...

Answer 2

Hint: for $r\in\mathbb{R}$, $$\{x\in X:f(x)>r\}=\bigcup_{n\in \mathbb{N}} \{x\in X:f(x)>q_n\}$$ where $(q_n)$ is a sequence of rational numbers decreasing to $r$.