$\chi_\mathbb{Q}(x) = 1$ if $x \in \mathbb{Q}, 0$ otherwise.
Well $\chi_\mathbb{Q}(x)$ is a measurable function if $\mathbb{Q}$ is a measurable set. $\mathbb{Q}$ is a measuable set under the Borel algebra. I.e. the algebra generated by the intervals.
And in this case $\mathbb{Q}$ is the union of an infinite number of points on the real line, each of which have measure $0$, so $\mathbb{Q}$ has measure $0$.
Is my understanding correct here?