## Proposition 7.3.12: Inverse Images are Measurable

If

*f*is a continuous real-valued function with a measurable set as domain, then the sets*f*,^{ -1}(a, b)*f*,^{ -1}(a, b]*f*, and^{ -1}[a, b)*f*are all measurable for any (extended) real numbers^{ -1}[a, b]*a < b*.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.