Interactive Real Analysis
- part of 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 -1[a, b), and
f -1[a, b]
are all measurable for any (extended) real numbers a < b.
Context
ContextTo 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.