Is it true that if E is measurable set and f is continuous on E, $B \subseteq \Bbb R$ a measurable set, then $f^{-1}(B)$ measurable?

Asked Oct 29 '19 at 03:06

Active Oct 29 '19 at 06:35

Viewed 74 times

Is it true that if E is measurable set and $f :E \to \Bbb R$ is continuous, $B \subseteq \Bbb R$ a measurable set, then $f^{-1}(B)$ is measurable set?

My guess: this is true.

Approach: Consider $\{A| f^{-1}(A) \text{is measurable}\}$ is a $\sigma$-algebra containing all open intervals.

Hence this is Borel sigma-algebra so $f^{-1}(B)$ is a Borel set and hence measurable.

As I got in the comment that the answer is not correct so please help me answering this question.

edited Oct 29 '19 at 06:35

asked Oct 29 '19 at 03:06

Ri-Li

9,038

By a measurable set do you mean a Borel set or a Lebesgue measurable set? – Kavi Rama Murthy Oct 29 '19 at 05:32
Lebesgue measurable set – Ri-Li Oct 29 '19 at 05:39
Your argument is not correct. If a sigma algebra contains all open intervals it contains all Borel sets but it need not contain Lebesgue measurable sets. – Kavi Rama Murthy Oct 29 '19 at 05:48
Then how to do that? – Ri-Li Oct 29 '19 at 06:33
https://math.stackexchange.com/questions/479441/example-of-a-continuous-function-that-is-not-lebesgue-measurable – Oct 29 '19 at 10:00

Is it true that if E is measurable set and f is continuous on E, $B \subseteq \Bbb R$ a measurable set, then $f^{-1}(B)$ measurable?

0 Answers0