An entropy inequality: $h_{mu}(beta,T)leq h_{mu}(alpha,T)+H_{mu}(beta|alpha)$
$begingroup$
Let T be a measure preserving transformation on the probability space $(X,mathcal{F},mu)$.
I have already solved this problem:
Suppose $alpha$ is a finite partition of $X$. Show that $h_{mu}(alpha, T) = h_{mu}(bigvee_{i=1}^n T^{-i}alpha,T)$
for any $ninmathbb{N}$.
But after a day I could not find a solution to the next one:
Let $alpha$ and $beta$ be finite partitions. Show that $h_{mu}(beta,T)leq h_{mu}(alpha,T)+H_{mu}(beta|alpha)$
I think it is undoable to do it by using the definition of entropies.
Since $h_{mu}(alpha,T)leq H_{mu}(alpha)$ and $H_{mu}(beta|alpha) = H_{mu}(alphaveebeta)-H_{mu}(alpha)$, it suffices to show
$H_{mu}(beta)+H_{mu}(alpha)leq h_{mu}(alpha,T)+H_{mu}(alphaveebeta)$
Maybe the previous problem should be used, but the next problem (c) says "use parts a and b to show...", so it is not very likely.
Is there anyone who can help? Thanks in advance.
measure-theory ergodic-theory entropy
asked Dec 31 '18 at 15:27
Rocco van VreumingenRocco van Vreumingen
928
$endgroup$
add a comment |
$begingroup$
Let T be a measure preserving transformation on the probability space $(X,mathcal{F},mu)$.
I have already solved this problem:
Suppose $alpha$ is a finite partition of $X$. Show that $h_{mu}(alpha, T) = h_{mu}(bigvee_{i=1}^n T^{-i}alpha,T)$
for any $ninmathbb{N}$.
But after a day I could not find a solution to the next one:
Let $alpha$ and $beta$ be finite partitions. Show that $h_{mu}(beta,T)leq h_{mu}(alpha,T)+H_{mu}(beta|alpha)$
I think it is undoable to do it by using the definition of entropies.
Since $h_{mu}(alpha,T)leq H_{mu}(alpha)$ and $H_{mu}(beta|alpha) = H_{mu}(alphaveebeta)-H_{mu}(alpha)$, it suffices to show
$H_{mu}(beta)+H_{mu}(alpha)leq h_{mu}(alpha,T)+H_{mu}(alphaveebeta)$
Maybe the previous problem should be used, but the next problem (c) says "use parts a and b to show...", so it is not very likely.
Is there anyone who can help? Thanks in advance.
measure-theory ergodic-theory entropy
asked Dec 31 '18 at 15:27
Rocco van VreumingenRocco van Vreumingen
928
$endgroup$
$begingroup$
No answer needed anymore because I already found the answer here: webusers.imj-prg.fr/~francois.le-maitre/gdt/KS.pdf
$endgroup$
– Rocco van Vreumingen
Jan 1 at 22:42
$begingroup$
delete question then
$endgroup$
– mathworker21
Jan 4 at 10:45
$begingroup$
@mathworker21 I'd rather not delete the question, because the answer in the link could also be useful for others.
$endgroup$
– Rocco van Vreumingen
Jan 7 at 17:26
add a comment |
$begingroup$
Let T be a measure preserving transformation on the probability space $(X,mathcal{F},mu)$.
I have already solved this problem:
Suppose $alpha$ is a finite partition of $X$. Show that $h_{mu}(alpha, T) = h_{mu}(bigvee_{i=1}^n T^{-i}alpha,T)$
for any $ninmathbb{N}$.
But after a day I could not find a solution to the next one:
Let $alpha$ and $beta$ be finite partitions. Show that $h_{mu}(beta,T)leq h_{mu}(alpha,T)+H_{mu}(beta|alpha)$
I think it is undoable to do it by using the definition of entropies.
Since $h_{mu}(alpha,T)leq H_{mu}(alpha)$ and $H_{mu}(beta|alpha) = H_{mu}(alphaveebeta)-H_{mu}(alpha)$, it suffices to show
$H_{mu}(beta)+H_{mu}(alpha)leq h_{mu}(alpha,T)+H_{mu}(alphaveebeta)$
Maybe the previous problem should be used, but the next problem (c) says "use parts a and b to show...", so it is not very likely.
Is there anyone who can help? Thanks in advance.
measure-theory ergodic-theory entropy
asked Dec 31 '18 at 15:27
Rocco van VreumingenRocco van Vreumingen
928
$endgroup$
Let T be a measure preserving transformation on the probability space $(X,mathcal{F},mu)$.
I have already solved this problem:
Suppose $alpha$ is a finite partition of $X$. Show that $h_{mu}(alpha, T) = h_{mu}(bigvee_{i=1}^n T^{-i}alpha,T)$
for any $ninmathbb{N}$.
But after a day I could not find a solution to the next one:
Let $alpha$ and $beta$ be finite partitions. Show that $h_{mu}(beta,T)leq h_{mu}(alpha,T)+H_{mu}(beta|alpha)$
I think it is undoable to do it by using the definition of entropies.
Since $h_{mu}(alpha,T)leq H_{mu}(alpha)$ and $H_{mu}(beta|alpha) = H_{mu}(alphaveebeta)-H_{mu}(alpha)$, it suffices to show
$H_{mu}(beta)+H_{mu}(alpha)leq h_{mu}(alpha,T)+H_{mu}(alphaveebeta)$
Maybe the previous problem should be used, but the next problem (c) says "use parts a and b to show...", so it is not very likely.
Is there anyone who can help? Thanks in advance.
measure-theory ergodic-theory entropy
measure-theory ergodic-theory entropy
asked Dec 31 '18 at 15:27
Rocco van VreumingenRocco van Vreumingen
928
asked Dec 31 '18 at 15:27
Rocco van VreumingenRocco van Vreumingen
928
asked Dec 31 '18 at 15:27
Rocco van VreumingenRocco van Vreumingen
928
asked Dec 31 '18 at 15:27
Rocco van VreumingenRocco van Vreumingen
928
asked Dec 31 '18 at 15:27
Rocco van VreumingenRocco van Vreumingen
928
928
$begingroup$
No answer needed anymore because I already found the answer here: webusers.imj-prg.fr/~francois.le-maitre/gdt/KS.pdf
$endgroup$
– Rocco van Vreumingen
Jan 1 at 22:42
$begingroup$
delete question then
$endgroup$
– mathworker21
Jan 4 at 10:45
$begingroup$
@mathworker21 I'd rather not delete the question, because the answer in the link could also be useful for others.
$endgroup$
– Rocco van Vreumingen
Jan 7 at 17:26
add a comment |
$begingroup$
No answer needed anymore because I already found the answer here: webusers.imj-prg.fr/~francois.le-maitre/gdt/KS.pdf
$endgroup$
– Rocco van Vreumingen
Jan 1 at 22:42
$begingroup$
delete question then
$endgroup$
– mathworker21
Jan 4 at 10:45
$begingroup$
@mathworker21 I'd rather not delete the question, because the answer in the link could also be useful for others.
$endgroup$
– Rocco van Vreumingen
Jan 7 at 17:26
$begingroup$
No answer needed anymore because I already found the answer here: webusers.imj-prg.fr/~francois.le-maitre/gdt/KS.pdf
$endgroup$
– Rocco van Vreumingen
Jan 1 at 22:42
$begingroup$
No answer needed anymore because I already found the answer here: webusers.imj-prg.fr/~francois.le-maitre/gdt/KS.pdf
$endgroup$
– Rocco van Vreumingen
Jan 1 at 22:42
$begingroup$
delete question then
$endgroup$
– mathworker21
Jan 4 at 10:45
$begingroup$
delete question then
$endgroup$
– mathworker21
Jan 4 at 10:45
$begingroup$
@mathworker21 I'd rather not delete the question, because the answer in the link could also be useful for others.
$endgroup$
– Rocco van Vreumingen
Jan 7 at 17:26
$begingroup$
@mathworker21 I'd rather not delete the question, because the answer in the link could also be useful for others.
$endgroup$
– Rocco van Vreumingen
Jan 7 at 17:26
add a comment |
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$begingroup$
No answer needed anymore because I already found the answer here: webusers.imj-prg.fr/~francois.le-maitre/gdt/KS.pdf
$endgroup$
– Rocco van Vreumingen
Jan 1 at 22:42
$begingroup$
delete question then
$endgroup$
– mathworker21
Jan 4 at 10:45
$begingroup$
@mathworker21 I'd rather not delete the question, because the answer in the link could also be useful for others.
$endgroup$
– Rocco van Vreumingen
Jan 7 at 17:26