If $(Ω,A,text P)$ is a non-atomic prob. space and $μ$ is a prob. measure on a Borel space there is a...
$begingroup$
Let
$(Omega,mathcal A,operatorname P)$ be a non-atomic probability space
$(E,mathcal E)$ be a Borel space
$mu$ be a probability measure on $(E,mathcal E)$
How can we show that there is a $(mathcal A,mathcal E)$-measurable $X:Omegato E$ with $X_astoperatorname P=mu$ (pushforward measure)?
I know that if $(mathbb R,mathcal B(mathbb R),nu)$ is a non-atomic probability space, the distribution function $F$ of $nu$ is continous and $F_astnu=mathcal U_{[0,:1]}$ (uniform distribution).
Now, I'll assume that for all $Binmathcal B(mathbb R)$, $(B,mathcal B(B))$ is isomorphic to $(mathbb R,mathcal B(mathbb R))$ (i.e. there is a measurable bijection between $(B,mathcal B(B))$ and $(mathbb R,mathcal B(mathbb R))$ with measurable inverse). Does anyone have a reference for that claim?
By that assumption, there is a $mathcal U_{[0,:1]}$-distributed random variable on any non-atomic probability space $(B,mathcal B(B),nu)$ with $Binmathcal B(mathbb R)$.
$(E,mathcal E)$ being Borel implies that $(E,mathcal E)$ is isomorphic to $(B,mathcal B(B))$.
How can we conclude? This answer seems to claim that the desired conclusion is possible, but I don't get how we need to argue exactly.
probability-theory measure-theory probability-distributions random-variables
asked Jan 3 at 21:39
0xbadf00d0xbadf00d
1,80641534
$endgroup$
add a comment |
$begingroup$
Let
$(Omega,mathcal A,operatorname P)$ be a non-atomic probability space
$(E,mathcal E)$ be a Borel space
$mu$ be a probability measure on $(E,mathcal E)$
How can we show that there is a $(mathcal A,mathcal E)$-measurable $X:Omegato E$ with $X_astoperatorname P=mu$ (pushforward measure)?
I know that if $(mathbb R,mathcal B(mathbb R),nu)$ is a non-atomic probability space, the distribution function $F$ of $nu$ is continous and $F_astnu=mathcal U_{[0,:1]}$ (uniform distribution).
Now, I'll assume that for all $Binmathcal B(mathbb R)$, $(B,mathcal B(B))$ is isomorphic to $(mathbb R,mathcal B(mathbb R))$ (i.e. there is a measurable bijection between $(B,mathcal B(B))$ and $(mathbb R,mathcal B(mathbb R))$ with measurable inverse). Does anyone have a reference for that claim?
By that assumption, there is a $mathcal U_{[0,:1]}$-distributed random variable on any non-atomic probability space $(B,mathcal B(B),nu)$ with $Binmathcal B(mathbb R)$.
$(E,mathcal E)$ being Borel implies that $(E,mathcal E)$ is isomorphic to $(B,mathcal B(B))$.
How can we conclude? This answer seems to claim that the desired conclusion is possible, but I don't get how we need to argue exactly.
probability-theory measure-theory probability-distributions random-variables
asked Jan 3 at 21:39
0xbadf00d0xbadf00d
1,80641534
$endgroup$
1
$begingroup$
See 'A course on Borel sets' by Srivastava and 'Measure Theory' by Cohen.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 23:55
$begingroup$
@KaviRamaMurthy Do you have a specific section in mind?
$endgroup$
– 0xbadf00d
Jan 4 at 0:17
$begingroup$
There are several isomorphism theorems in measure theory and the they are quite deep. You will, have to spend quiet a bit of time to digest these. I suggest you first look at statements of some of these theorems from Cohen's book first.
$endgroup$
– Kavi Rama Murthy
Jan 4 at 0:21
$begingroup$
@KaviRamaMurthy Your comment is only related to the isomorphism between $(mathbb R,mathcal B(mathbb R))$ and $(B,mathcal B(B))$ I've asked for, right? If so, do you have any idea how we need to proceed once this existence is known?
$endgroup$
– 0xbadf00d
Jan 4 at 0:23
add a comment |
$begingroup$
Let
$(Omega,mathcal A,operatorname P)$ be a non-atomic probability space
$(E,mathcal E)$ be a Borel space
$mu$ be a probability measure on $(E,mathcal E)$
How can we show that there is a $(mathcal A,mathcal E)$-measurable $X:Omegato E$ with $X_astoperatorname P=mu$ (pushforward measure)?
I know that if $(mathbb R,mathcal B(mathbb R),nu)$ is a non-atomic probability space, the distribution function $F$ of $nu$ is continous and $F_astnu=mathcal U_{[0,:1]}$ (uniform distribution).
Now, I'll assume that for all $Binmathcal B(mathbb R)$, $(B,mathcal B(B))$ is isomorphic to $(mathbb R,mathcal B(mathbb R))$ (i.e. there is a measurable bijection between $(B,mathcal B(B))$ and $(mathbb R,mathcal B(mathbb R))$ with measurable inverse). Does anyone have a reference for that claim?
By that assumption, there is a $mathcal U_{[0,:1]}$-distributed random variable on any non-atomic probability space $(B,mathcal B(B),nu)$ with $Binmathcal B(mathbb R)$.
$(E,mathcal E)$ being Borel implies that $(E,mathcal E)$ is isomorphic to $(B,mathcal B(B))$.
How can we conclude? This answer seems to claim that the desired conclusion is possible, but I don't get how we need to argue exactly.
probability-theory measure-theory probability-distributions random-variables
asked Jan 3 at 21:39
0xbadf00d0xbadf00d
1,80641534
$endgroup$
Let
$(Omega,mathcal A,operatorname P)$ be a non-atomic probability space
$(E,mathcal E)$ be a Borel space
$mu$ be a probability measure on $(E,mathcal E)$
How can we show that there is a $(mathcal A,mathcal E)$-measurable $X:Omegato E$ with $X_astoperatorname P=mu$ (pushforward measure)?
I know that if $(mathbb R,mathcal B(mathbb R),nu)$ is a non-atomic probability space, the distribution function $F$ of $nu$ is continous and $F_astnu=mathcal U_{[0,:1]}$ (uniform distribution).
Now, I'll assume that for all $Binmathcal B(mathbb R)$, $(B,mathcal B(B))$ is isomorphic to $(mathbb R,mathcal B(mathbb R))$ (i.e. there is a measurable bijection between $(B,mathcal B(B))$ and $(mathbb R,mathcal B(mathbb R))$ with measurable inverse). Does anyone have a reference for that claim?
By that assumption, there is a $mathcal U_{[0,:1]}$-distributed random variable on any non-atomic probability space $(B,mathcal B(B),nu)$ with $Binmathcal B(mathbb R)$.
$(E,mathcal E)$ being Borel implies that $(E,mathcal E)$ is isomorphic to $(B,mathcal B(B))$.
How can we conclude? This answer seems to claim that the desired conclusion is possible, but I don't get how we need to argue exactly.
probability-theory measure-theory probability-distributions random-variables
probability-theory measure-theory probability-distributions random-variables
asked Jan 3 at 21:39
0xbadf00d0xbadf00d
1,80641534
asked Jan 3 at 21:39
0xbadf00d0xbadf00d
1,80641534
edited Jan 4 at 0:19
0xbadf00d
asked Jan 3 at 21:39
0xbadf00d0xbadf00d
1,80641534
asked Jan 3 at 21:39
0xbadf00d0xbadf00d
1,80641534
asked Jan 3 at 21:39
0xbadf00d0xbadf00d
1,80641534
1,80641534
1
$begingroup$
See 'A course on Borel sets' by Srivastava and 'Measure Theory' by Cohen.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 23:55
$begingroup$
@KaviRamaMurthy Do you have a specific section in mind?
$endgroup$
– 0xbadf00d
Jan 4 at 0:17
$begingroup$
There are several isomorphism theorems in measure theory and the they are quite deep. You will, have to spend quiet a bit of time to digest these. I suggest you first look at statements of some of these theorems from Cohen's book first.
$endgroup$
– Kavi Rama Murthy
Jan 4 at 0:21
$begingroup$
@KaviRamaMurthy Your comment is only related to the isomorphism between $(mathbb R,mathcal B(mathbb R))$ and $(B,mathcal B(B))$ I've asked for, right? If so, do you have any idea how we need to proceed once this existence is known?
$endgroup$
– 0xbadf00d
Jan 4 at 0:23
add a comment |
1
$begingroup$
See 'A course on Borel sets' by Srivastava and 'Measure Theory' by Cohen.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 23:55
$begingroup$
@KaviRamaMurthy Do you have a specific section in mind?
$endgroup$
– 0xbadf00d
Jan 4 at 0:17
$begingroup$
There are several isomorphism theorems in measure theory and the they are quite deep. You will, have to spend quiet a bit of time to digest these. I suggest you first look at statements of some of these theorems from Cohen's book first.
$endgroup$
– Kavi Rama Murthy
Jan 4 at 0:21
$begingroup$
@KaviRamaMurthy Your comment is only related to the isomorphism between $(mathbb R,mathcal B(mathbb R))$ and $(B,mathcal B(B))$ I've asked for, right? If so, do you have any idea how we need to proceed once this existence is known?
$endgroup$
– 0xbadf00d
Jan 4 at 0:23
1
1
$begingroup$
See 'A course on Borel sets' by Srivastava and 'Measure Theory' by Cohen.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 23:55
$begingroup$
See 'A course on Borel sets' by Srivastava and 'Measure Theory' by Cohen.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 23:55
$begingroup$
@KaviRamaMurthy Do you have a specific section in mind?
$endgroup$
– 0xbadf00d
Jan 4 at 0:17
$begingroup$
@KaviRamaMurthy Do you have a specific section in mind?
$endgroup$
– 0xbadf00d
Jan 4 at 0:17
$begingroup$
There are several isomorphism theorems in measure theory and the they are quite deep. You will, have to spend quiet a bit of time to digest these. I suggest you first look at statements of some of these theorems from Cohen's book first.
$endgroup$
– Kavi Rama Murthy
Jan 4 at 0:21
$begingroup$
There are several isomorphism theorems in measure theory and the they are quite deep. You will, have to spend quiet a bit of time to digest these. I suggest you first look at statements of some of these theorems from Cohen's book first.
$endgroup$
– Kavi Rama Murthy
Jan 4 at 0:21
$begingroup$
@KaviRamaMurthy Your comment is only related to the isomorphism between $(mathbb R,mathcal B(mathbb R))$ and $(B,mathcal B(B))$ I've asked for, right? If so, do you have any idea how we need to proceed once this existence is known?
$endgroup$
– 0xbadf00d
Jan 4 at 0:23
$begingroup$
@KaviRamaMurthy Your comment is only related to the isomorphism between $(mathbb R,mathcal B(mathbb R))$ and $(B,mathcal B(B))$ I've asked for, right? If so, do you have any idea how we need to proceed once this existence is known?
$endgroup$
– 0xbadf00d
Jan 4 at 0:23
add a comment |
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1
$begingroup$
See 'A course on Borel sets' by Srivastava and 'Measure Theory' by Cohen.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 23:55
$begingroup$
@KaviRamaMurthy Do you have a specific section in mind?
$endgroup$
– 0xbadf00d
Jan 4 at 0:17
$begingroup$
There are several isomorphism theorems in measure theory and the they are quite deep. You will, have to spend quiet a bit of time to digest these. I suggest you first look at statements of some of these theorems from Cohen's book first.
$endgroup$
– Kavi Rama Murthy
Jan 4 at 0:21
$begingroup$
@KaviRamaMurthy Your comment is only related to the isomorphism between $(mathbb R,mathcal B(mathbb R))$ and $(B,mathcal B(B))$ I've asked for, right? If so, do you have any idea how we need to proceed once this existence is known?
$endgroup$
– 0xbadf00d
Jan 4 at 0:23