Tips to show that the Random Variable $X$ is $\mathcal{B}(\mathbb R)-\mathcal{B}(\mathbb R)-$measurable

Question

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?

Answer 1

Hint using your first approach:

Let $A \in \mathcal{B}(\mathbb{R})$, then consider the two cases: $0 \in A$ and $0 \notin A$. Now try using that all closed and all open subsets of $\mathbb{R}$ are contained in $\mathcal{B}(\mathbb{R})$ and the definition of a $\sigma$-algebra (in particular, that it is closed under countable unions).