Timeline for Preimage of null sets under a monotone increasing function
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jan 24 at 15:34 | vote | accept | Julian | ||
| Jan 24 at 13:31 | answer | added | Julian | timeline score: 2 | |
| Jan 19 at 23:01 | comment | added | Gro-Tsen | A simple and standard counterexample: let $C'$ be a fat Cantor set obtained by removing the middle $1/2^n$ at each stage, and $C$ be the standard (null) Cantor set obtained by removing the middle third at each stage, both in $I=[0,1]$. Construct $f\colon I\to I$ by taking the increasing bijective affine map from each open interval component of the complement of $C'$ to the corresponding one of $C$, and extend it by continuity to $I$. Then $f\colon I\to I$ is a continuous increasing bijection and $f^{-1}(C) = C'$, yet $C$ is null while $C'$ is not. | |
| Jan 19 at 20:25 | comment | added | Christian Remling | @IosifPinelis: This is the situation I had in mind: $g=f^{-1}$ a strictly increasing, singular continuous (as in singular + continuous) function, so that $f$ will also be strictly increasing and continuous. | |
| Jan 19 at 19:45 | answer | added | Ben Johnsrude | timeline score: 3 | |
| Jan 19 at 13:53 | history | edited | Julian | CC BY-SA 4.0 |
Added additional assumption
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| Jan 19 at 1:51 | comment | added | Iosif Pinelis | @ChristianRemling : Do you know the answer if we additionally require that $f$ be continuous? | |
| Jan 19 at 0:57 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Jan 18 at 22:18 | review | Close votes | |||
| Jan 27 at 3:05 | |||||
| Jan 18 at 22:01 | comment | added | Christian Remling | No. If $f$ is strictly increasing, then you are asking if $g=f^{-1}$ maps null sets to null sets, which is false for singular continuous functions. | |
| Jan 18 at 19:28 | history | asked | Julian | CC BY-SA 4.0 |