Timeline for Existence of ergodic measure for measurable maps
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 14, 2013 at 20:05 | comment | added | Vaughn Climenhaga | Yes, if $T$ is a measurable transformation and $\mu$ is a $T$-invariant measure, then there must exist an ergodic measure. This is using the ergodic decomposition, see for example this question: mathoverflow.net/q/124066/5701 | |
| Aug 14, 2013 at 19:45 | comment | added | John Learner | Great, thanks. But would then a sufficient condition be the existence of an invariant measure? | |
| Aug 14, 2013 at 19:25 | answer | added | Vaughn Climenhaga | timeline score: 5 | |
| Aug 14, 2013 at 19:07 | comment | added | Vaughn Climenhaga | $x\mapsto x+1$ in $\mathbb{R}$ is the standard counterexample, if you want $\Omega$ to be compact you can take $[0,1]$ with $T\colon x\mapsto x/2$ for $x>0$ and $T(0)=1$. | |
| Aug 14, 2013 at 19:06 | comment | added | Vaughn Climenhaga | Measurability alone is not enough to guarantee existence of invariant (or ergodic) measures. See math.stackexchange.com/questions/94981/… and mathoverflow.net/questions/66669/… | |
| Aug 14, 2013 at 18:55 | review | First posts | |||
| Aug 14, 2013 at 18:57 | |||||
| Aug 14, 2013 at 18:37 | history | asked | John Learner | CC BY-SA 3.0 |