7.3. Measures

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.

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.

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