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Small question in a proposition involving Ergodicity
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Ergodicity -- A measure preserving transformation T on the space (X, $mathcalB$, $mu$) is called Ergodic iff $forall$ B $inmathcalB$ satisfying T$^-1$B = B we have $mu$(B) = 0 or 1.
Let T be a measure preserving transformation of a space (X, $mathcalB$, $mu$). Then the following are equivalent:
T is ergodic;
For all f $in$ L$^1$ (X, $mathcalB$, $mu$) satisfying f $circ$ T = f a.e. then f is constant a.e.
The book I am reading says that "we can replace L$^1$ in above proposition by measurable or L$^2". Why is that true? Can anyone help me on this?
lp-spaces ergodic-theory
asked 55 mins ago
HumbleStudent
628311
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favorite
Ergodicity -- A measure preserving transformation T on the space (X, $mathcalB$, $mu$) is called Ergodic iff $forall$ B $inmathcalB$ satisfying T$^-1$B = B we have $mu$(B) = 0 or 1.
Let T be a measure preserving transformation of a space (X, $mathcalB$, $mu$). Then the following are equivalent:
T is ergodic;
For all f $in$ L$^1$ (X, $mathcalB$, $mu$) satisfying f $circ$ T = f a.e. then f is constant a.e.
The book I am reading says that "we can replace L$^1$ in above proposition by measurable or L$^2". Why is that true? Can anyone help me on this?
lp-spaces ergodic-theory
asked 55 mins ago
HumbleStudent
628311
Have you seen the proof of $1. iff 2.$? If so, look at the proof, you should see that it doesn't matter whether $f $ belongs to $L^1$ or $L^2$. i.e the same proof applies.
– Zestylemonzi
40 mins ago
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Ergodicity -- A measure preserving transformation T on the space (X, $mathcalB$, $mu$) is called Ergodic iff $forall$ B $inmathcalB$ satisfying T$^-1$B = B we have $mu$(B) = 0 or 1.
Let T be a measure preserving transformation of a space (X, $mathcalB$, $mu$). Then the following are equivalent:
T is ergodic;
For all f $in$ L$^1$ (X, $mathcalB$, $mu$) satisfying f $circ$ T = f a.e. then f is constant a.e.
The book I am reading says that "we can replace L$^1$ in above proposition by measurable or L$^2". Why is that true? Can anyone help me on this?
lp-spaces ergodic-theory
asked 55 mins ago
HumbleStudent
628311
Ergodicity -- A measure preserving transformation T on the space (X, $mathcalB$, $mu$) is called Ergodic iff $forall$ B $inmathcalB$ satisfying T$^-1$B = B we have $mu$(B) = 0 or 1.
Let T be a measure preserving transformation of a space (X, $mathcalB$, $mu$). Then the following are equivalent:
T is ergodic;
For all f $in$ L$^1$ (X, $mathcalB$, $mu$) satisfying f $circ$ T = f a.e. then f is constant a.e.
The book I am reading says that "we can replace L$^1$ in above proposition by measurable or L$^2". Why is that true? Can anyone help me on this?
lp-spaces ergodic-theory
asked 55 mins ago
HumbleStudent
628311
asked 55 mins ago
HumbleStudent
628311
asked 55 mins ago
HumbleStudent
628311
asked 55 mins ago
HumbleStudent
628311
628311
Have you seen the proof of $1. iff 2.$? If so, look at the proof, you should see that it doesn't matter whether $f $ belongs to $L^1$ or $L^2$. i.e the same proof applies.
– Zestylemonzi
40 mins ago
add a comment |Â
Have you seen the proof of $1. iff 2.$? If so, look at the proof, you should see that it doesn't matter whether $f $ belongs to $L^1$ or $L^2$. i.e the same proof applies.
– Zestylemonzi
40 mins ago
Have you seen the proof of $1. iff 2.$? If so, look at the proof, you should see that it doesn't matter whether $f $ belongs to $L^1$ or $L^2$. i.e the same proof applies.
– Zestylemonzi
40 mins ago
Have you seen the proof of $1. iff 2.$? If so, look at the proof, you should see that it doesn't matter whether $f $ belongs to $L^1$ or $L^2$. i.e the same proof applies.
– Zestylemonzi
40 mins ago
add a comment |Â
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Have you seen the proof of $1. iff 2.$? If so, look at the proof, you should see that it doesn't matter whether $f $ belongs to $L^1$ or $L^2$. i.e the same proof applies.
– Zestylemonzi
40 mins ago