Tips to show that the Random Variable $X$ is $mathcal{B}(mathbb R)-mathcal{B}(mathbb R)-$measurable
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Let $X: mathbb R to mathbb R$ , where $forall omega in mathbb R -mathbb Q: X(omega)=0$ . Show $X$ is $mathcal{B}(mathbb R)-mathcal{B}(mathbb R)-$ measurable function: My ideas: Using the definition of measurability I could take any $C in mathcal{B}(mathbb R)$ and I then want to show $X^{-1}(C) in mathcal{B}(mathbb R)$ , but from the definition of the $X$ , all I can take away is that $mathbb R - mathbb Q subseteq X^{-1}({0})$ but this in no way shows measurability. Another approach could be to take a generator $mathcal{E}$ , and prove $X^{-1}(mathcal{E}) subseteq mathcal{B}(mathbb R)$ . I think the biggest problem is that I have no definition of $X(omega)$ where $omega in mathbb Q$ . Any tips?
probability random-variables borel-sets borel-measures