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Showing that transition is measurable

1

1

$begingroup$

Let $P:mathbbR^ntimes mathscrB_mathbbR^n rightarrow [0,infty]$ be a function such that $P(x,-)$ is a probability measure for each $x$ and $P(-,A)$ is (borel) measurable for each $A$.

Let $Ain mathscrB_mathbbR^notimes mathscrB_mathbbR^n$. Then, how do I prove that the map $xmapsto P(x,A_x)$ is measurable? ($A_x$ denotes the $x$-section of $A$)

real-analysis measure-theory measurable-functions measurable-sets

share|cite|improve this question

edited Mar 31 at 23:32

J. W. Tanner

4,8071420

asked Mar 31 at 23:11

RubertosRubertos

5,7812826

$endgroup$

add a comment | 

1

1

$begingroup$

Let $P:mathbbR^ntimes mathscrB_mathbbR^n rightarrow [0,infty]$ be a function such that $P(x,-)$ is a probability measure for each $x$ and $P(-,A)$ is (borel) measurable for each $A$.

Let $Ain mathscrB_mathbbR^notimes mathscrB_mathbbR^n$. Then, how do I prove that the map $xmapsto P(x,A_x)$ is measurable? ($A_x$ denotes the $x$-section of $A$)

real-analysis measure-theory measurable-functions measurable-sets

share|cite|improve this question

edited Mar 31 at 23:32

J. W. Tanner

4,8071420

asked Mar 31 at 23:11

RubertosRubertos

5,7812826

$endgroup$

add a comment | 

$begingroup$

Let $P:mathbbR^ntimes mathscrB_mathbbR^n rightarrow [0,infty]$ be a function such that $P(x,-)$ is a probability measure for each $x$ and $P(-,A)$ is (borel) measurable for each $A$.

Let $Ain mathscrB_mathbbR^notimes mathscrB_mathbbR^n$. Then, how do I prove that the map $xmapsto P(x,A_x)$ is measurable? ($A_x$ denotes the $x$-section of $A$)

real-analysis measure-theory measurable-functions measurable-sets

share|cite|improve this question

edited Mar 31 at 23:32

J. W. Tanner

4,8071420

asked Mar 31 at 23:11

RubertosRubertos

5,7812826

$endgroup$

share|cite|improve this question

edited Mar 31 at 23:32

J. W. Tanner

4,8071420

asked Mar 31 at 23:11

RubertosRubertos

5,7812826

share|cite|improve this question

edited Mar 31 at 23:32

J. W. Tanner

4,8071420

asked Mar 31 at 23:11

RubertosRubertos

5,7812826

edited Mar 31 at 23:32

J. W. Tanner

4,8071420

edited Mar 31 at 23:32

J. W. Tanner

4,8071420

asked Mar 31 at 23:11

RubertosRubertos

5,7812826

asked Mar 31 at 23:11

RubertosRubertos

5,7812826

1 Answer
1

active

oldest

votes

2

$begingroup$

Let $A$ and $B$ be Borel sets. Then $P(x,(Atimes B)_x)=P(x,B)$ if $x in A$ and $0$ if $x notin A$, so $P(x,(Atimes B)_x)$ is measurable. If $E$ is a finite disjoint union of measurable rectangles $Atimes B$ then $P(x,E_x)$ is a finite sum of functions of above type so it is measurable. The class of all finite disjoint union of measurable rectangles is an algebra which generates the product sigma algebra; also the class of all sets $E$ in the product sigma algebra such that $P(x,E_x)$ is measurable is a monotone class. Use Monotone Class Theorem to complete the proof.

share|cite|improve this answer

answered Mar 31 at 23:19

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

$endgroup$

add a comment | 

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2

$begingroup$

Let $A$ and $B$ be Borel sets. Then $P(x,(Atimes B)_x)=P(x,B)$ if $x in A$ and $0$ if $x notin A$, so $P(x,(Atimes B)_x)$ is measurable. If $E$ is a finite disjoint union of measurable rectangles $Atimes B$ then $P(x,E_x)$ is a finite sum of functions of above type so it is measurable. The class of all finite disjoint union of measurable rectangles is an algebra which generates the product sigma algebra; also the class of all sets $E$ in the product sigma algebra such that $P(x,E_x)$ is measurable is a monotone class. Use Monotone Class Theorem to complete the proof.

share|cite|improve this answer

answered Mar 31 at 23:19

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

$endgroup$

add a comment | 

2

$begingroup$

Let $A$ and $B$ be Borel sets. Then $P(x,(Atimes B)_x)=P(x,B)$ if $x in A$ and $0$ if $x notin A$, so $P(x,(Atimes B)_x)$ is measurable. If $E$ is a finite disjoint union of measurable rectangles $Atimes B$ then $P(x,E_x)$ is a finite sum of functions of above type so it is measurable. The class of all finite disjoint union of measurable rectangles is an algebra which generates the product sigma algebra; also the class of all sets $E$ in the product sigma algebra such that $P(x,E_x)$ is measurable is a monotone class. Use Monotone Class Theorem to complete the proof.

share|cite|improve this answer

answered Mar 31 at 23:19

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

$endgroup$

add a comment | 

$begingroup$

Let $A$ and $B$ be Borel sets. Then $P(x,(Atimes B)_x)=P(x,B)$ if $x in A$ and $0$ if $x notin A$, so $P(x,(Atimes B)_x)$ is measurable. If $E$ is a finite disjoint union of measurable rectangles $Atimes B$ then $P(x,E_x)$ is a finite sum of functions of above type so it is measurable. The class of all finite disjoint union of measurable rectangles is an algebra which generates the product sigma algebra; also the class of all sets $E$ in the product sigma algebra such that $P(x,E_x)$ is measurable is a monotone class. Use Monotone Class Theorem to complete the proof.

share|cite|improve this answer

answered Mar 31 at 23:19

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

$endgroup$

share|cite|improve this answer

answered Mar 31 at 23:19

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

answered Mar 31 at 23:19

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

answered Mar 31 at 23:19

Kavi Rama MurthyKavi Rama Murthy

74.9k53270