Proposition 7.3.12: Inverse Images are Measurable
To prove this we will use a result from the somewhat obscure section on continuity and topology. In particular, we showed in that section that a function is continuous if and only the inverse image of every open set is open. Since open sets are measurable, it shows that f -1(a, b) is measurable for f continuous. The same is true for the inverse image of closed sets.
The remaining inverse images of the half open intervals are ... what else, left as exercise.