Mixing
Could anyone help me to prove the followings are equivalent?
$begingroup$
$Ksubseteq mathbb R^n$ be closed then show that TFAE
A probability measure $mu$ on $K$ is ergodic.
(1) Every $mu$-invariant set of $mu$ measure zero or one.
(2) $mu$ can-not be decomposed as $mu=xmu_1+(1-x)mu_2$ with $xin (0,1)$ for two invariant measures $mu_1,mu_2$
Thanks for helping me understanding and proving the above.
measure-theory ergodic-theory
asked Jan 30 at 16:44
miosakimiosaki
2,43411537
$endgroup$
add a comment |
$begingroup$
$Ksubseteq mathbb R^n$ be closed then show that TFAE
A probability measure $mu$ on $K$ is ergodic.
(1) Every $mu$-invariant set of $mu$ measure zero or one.
(2) $mu$ can-not be decomposed as $mu=xmu_1+(1-x)mu_2$ with $xin (0,1)$ for two invariant measures $mu_1,mu_2$
Thanks for helping me understanding and proving the above.
measure-theory ergodic-theory
asked Jan 30 at 16:44
miosakimiosaki
2,43411537
$endgroup$
3
$begingroup$
What is your definition of ergodic measure? Because (2) is one of the possible definitions.
$endgroup$
– p4sch
Jan 30 at 17:13
add a comment |
$begingroup$
$Ksubseteq mathbb R^n$ be closed then show that TFAE
A probability measure $mu$ on $K$ is ergodic.
(1) Every $mu$-invariant set of $mu$ measure zero or one.
(2) $mu$ can-not be decomposed as $mu=xmu_1+(1-x)mu_2$ with $xin (0,1)$ for two invariant measures $mu_1,mu_2$
Thanks for helping me understanding and proving the above.
measure-theory ergodic-theory
asked Jan 30 at 16:44
miosakimiosaki
2,43411537
$endgroup$
$Ksubseteq mathbb R^n$ be closed then show that TFAE
A probability measure $mu$ on $K$ is ergodic.
(1) Every $mu$-invariant set of $mu$ measure zero or one.
(2) $mu$ can-not be decomposed as $mu=xmu_1+(1-x)mu_2$ with $xin (0,1)$ for two invariant measures $mu_1,mu_2$
Thanks for helping me understanding and proving the above.
measure-theory ergodic-theory
measure-theory ergodic-theory
asked Jan 30 at 16:44
miosakimiosaki
2,43411537
asked Jan 30 at 16:44
miosakimiosaki
2,43411537
edited Jan 31 at 11:56
miosaki
asked Jan 30 at 16:44
miosakimiosaki
2,43411537
asked Jan 30 at 16:44
miosakimiosaki
2,43411537
asked Jan 30 at 16:44
miosakimiosaki
2,43411537
2,43411537
3
$begingroup$
What is your definition of ergodic measure? Because (2) is one of the possible definitions.
$endgroup$
– p4sch
Jan 30 at 17:13
add a comment |
3
$begingroup$
What is your definition of ergodic measure? Because (2) is one of the possible definitions.
$endgroup$
– p4sch
Jan 30 at 17:13
3
3
$begingroup$
What is your definition of ergodic measure? Because (2) is one of the possible definitions.
$endgroup$
– p4sch
Jan 30 at 17:13
$begingroup$
What is your definition of ergodic measure? Because (2) is one of the possible definitions.
$endgroup$
– p4sch
Jan 30 at 17:13
add a comment |
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$begingroup$
What is your definition of ergodic measure? Because (2) is one of the possible definitions.
$endgroup$
– p4sch
Jan 30 at 17:13