Examples of the Carathéodory extension theorem

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What are examples of the Carathéodory Extension Theorem that is not Lebesgue measure or Hausdorff measure?

real-analysis analysis measure-theory

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edited Aug 2 at 18:31

asked Aug 2 at 18:23

user109871

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What are examples of the Carathéodory Extension Theorem that is not Lebesgue measure or Hausdorff measure?

real-analysis analysis measure-theory

share|cite|improve this question

edited Aug 2 at 18:31

asked Aug 2 at 18:23

user109871

507217

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

What are examples of the Carathéodory Extension Theorem that is not Lebesgue measure or Hausdorff measure?

real-analysis analysis measure-theory

share|cite|improve this question

edited Aug 2 at 18:31

asked Aug 2 at 18:23

user109871

507217

What are examples of the Carathéodory Extension Theorem that is not Lebesgue measure or Hausdorff measure?

real-analysis analysis measure-theory

share|cite|improve this question

edited Aug 2 at 18:31

asked Aug 2 at 18:23

user109871

507217

share|cite|improve this question

share|cite|improve this question

edited Aug 2 at 18:31

edited Aug 2 at 18:31

edited Aug 2 at 18:31

asked Aug 2 at 18:23

user109871

507217

asked Aug 2 at 18:23

user109871

507217

asked Aug 2 at 18:23

user109871

507217

507217

add a comment | 

add a comment | 

2 Answers
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One very important example is the construction of product measures. Given measures $mu$ and $nu$ on measurable spaces $X$ and $Y$, we seek to define a product measure $mutimes nu$ on $Xtimes Y$. Now, it is easy to define what this product measure should be on rectangles: if $Asubseteq X$ and $Bsubseteq Y$ are measurable, then $(mutimesnu)(Atimes B)$ should be $mu(A)nu(B)$ (just like the usual formula for the area of a rectangle in the plane). This can then be extended to finite unions of rectangles, and then the Caratheodory extension theorem can be used to extend it to a measure defined on the entire $sigma$-algebra generated by such rectangles.

share|cite|improve this answer

answered Aug 2 at 18:41

Eric Wofsey

161k12188297

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Do you know of a useful product of measure spaces that are not euclidean spaces, where the product measure is used?
  – user109871
  Aug 2 at 18:47
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Sure, it's used all the time in probability theory. Product measures are how you rigorously model something like "independent samples": you take a product of copies of your probability space, one for each sample.
  – Eric Wofsey
  Aug 2 at 18:53
  

  
  
  
  

add a comment | 

up vote
2
down vote

The concept of a cumulative distribution functions (cdf) is closely related to Carathéodory Extension Theorem. The construction is fairly similar to the Lebesgue measure though.

Let $P$ be a probability measure on $(mathbbR, mathcalBmathbbR)$. The cdf of $P$ is the function $$F: mathbbR rightarrow [0,1], x mapsto P((-infty, x]).$$ Given the measure $P$, one can obviously construct $F$, but $F$ can also be used to reconstruct $P$:

Given a cdf $F$. The according probability measure $P$ is the unique measure on $(mathbbR, mathcalBmathbbR)$, such that $P((a,b]) = F(b)-F(a)$. One can prove that $P$ exists and that it is unique, using the Carathéodory Extension Theorem.

share|cite|improve this answer

answered Aug 2 at 18:44

Jonas

259210

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
3
down vote

One very important example is the construction of product measures. Given measures $mu$ and $nu$ on measurable spaces $X$ and $Y$, we seek to define a product measure $mutimes nu$ on $Xtimes Y$. Now, it is easy to define what this product measure should be on rectangles: if $Asubseteq X$ and $Bsubseteq Y$ are measurable, then $(mutimesnu)(Atimes B)$ should be $mu(A)nu(B)$ (just like the usual formula for the area of a rectangle in the plane). This can then be extended to finite unions of rectangles, and then the Caratheodory extension theorem can be used to extend it to a measure defined on the entire $sigma$-algebra generated by such rectangles.

share|cite|improve this answer

answered Aug 2 at 18:41

Eric Wofsey

161k12188297

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Do you know of a useful product of measure spaces that are not euclidean spaces, where the product measure is used?
  – user109871
  Aug 2 at 18:47
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Sure, it's used all the time in probability theory. Product measures are how you rigorously model something like "independent samples": you take a product of copies of your probability space, one for each sample.
  – Eric Wofsey
  Aug 2 at 18:53
  

  
  
  
  

add a comment | 

up vote
3
down vote

One very important example is the construction of product measures. Given measures $mu$ and $nu$ on measurable spaces $X$ and $Y$, we seek to define a product measure $mutimes nu$ on $Xtimes Y$. Now, it is easy to define what this product measure should be on rectangles: if $Asubseteq X$ and $Bsubseteq Y$ are measurable, then $(mutimesnu)(Atimes B)$ should be $mu(A)nu(B)$ (just like the usual formula for the area of a rectangle in the plane). This can then be extended to finite unions of rectangles, and then the Caratheodory extension theorem can be used to extend it to a measure defined on the entire $sigma$-algebra generated by such rectangles.

share|cite|improve this answer

answered Aug 2 at 18:41

Eric Wofsey

161k12188297

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Do you know of a useful product of measure spaces that are not euclidean spaces, where the product measure is used?
  – user109871
  Aug 2 at 18:47
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Sure, it's used all the time in probability theory. Product measures are how you rigorously model something like "independent samples": you take a product of copies of your probability space, one for each sample.
  – Eric Wofsey
  Aug 2 at 18:53
  

  
  
  
  

add a comment | 

up vote
3
down vote

up vote
3
down vote

One very important example is the construction of product measures. Given measures $mu$ and $nu$ on measurable spaces $X$ and $Y$, we seek to define a product measure $mutimes nu$ on $Xtimes Y$. Now, it is easy to define what this product measure should be on rectangles: if $Asubseteq X$ and $Bsubseteq Y$ are measurable, then $(mutimesnu)(Atimes B)$ should be $mu(A)nu(B)$ (just like the usual formula for the area of a rectangle in the plane). This can then be extended to finite unions of rectangles, and then the Caratheodory extension theorem can be used to extend it to a measure defined on the entire $sigma$-algebra generated by such rectangles.

share|cite|improve this answer

answered Aug 2 at 18:41

Eric Wofsey

161k12188297

One very important example is the construction of product measures. Given measures $mu$ and $nu$ on measurable spaces $X$ and $Y$, we seek to define a product measure $mutimes nu$ on $Xtimes Y$. Now, it is easy to define what this product measure should be on rectangles: if $Asubseteq X$ and $Bsubseteq Y$ are measurable, then $(mutimesnu)(Atimes B)$ should be $mu(A)nu(B)$ (just like the usual formula for the area of a rectangle in the plane). This can then be extended to finite unions of rectangles, and then the Caratheodory extension theorem can be used to extend it to a measure defined on the entire $sigma$-algebra generated by such rectangles.

share|cite|improve this answer

answered Aug 2 at 18:41

Eric Wofsey

161k12188297

share|cite|improve this answer

share|cite|improve this answer

answered Aug 2 at 18:41

Eric Wofsey

161k12188297

answered Aug 2 at 18:41

Eric Wofsey

161k12188297

answered Aug 2 at 18:41

Eric Wofsey

161k12188297

161k12188297

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Do you know of a useful product of measure spaces that are not euclidean spaces, where the product measure is used?
  – user109871
  Aug 2 at 18:47
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Sure, it's used all the time in probability theory. Product measures are how you rigorously model something like "independent samples": you take a product of copies of your probability space, one for each sample.
  – Eric Wofsey
  Aug 2 at 18:53
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Do you know of a useful product of measure spaces that are not euclidean spaces, where the product measure is used?
  – user109871
  Aug 2 at 18:47
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Sure, it's used all the time in probability theory. Product measures are how you rigorously model something like "independent samples": you take a product of copies of your probability space, one for each sample.
  – Eric Wofsey
  Aug 2 at 18:53
  

  
  
  
  

Do you know of a useful product of measure spaces that are not euclidean spaces, where the product measure is used?
– user109871
Aug 2 at 18:47

Do you know of a useful product of measure spaces that are not euclidean spaces, where the product measure is used?
– user109871
Aug 2 at 18:47

1

1

Sure, it's used all the time in probability theory. Product measures are how you rigorously model something like "independent samples": you take a product of copies of your probability space, one for each sample.
– Eric Wofsey
Aug 2 at 18:53

Sure, it's used all the time in probability theory. Product measures are how you rigorously model something like "independent samples": you take a product of copies of your probability space, one for each sample.
– Eric Wofsey
Aug 2 at 18:53

add a comment | 

up vote
2
down vote

The concept of a cumulative distribution functions (cdf) is closely related to Carathéodory Extension Theorem. The construction is fairly similar to the Lebesgue measure though.

Let $P$ be a probability measure on $(mathbbR, mathcalBmathbbR)$. The cdf of $P$ is the function $$F: mathbbR rightarrow [0,1], x mapsto P((-infty, x]).$$ Given the measure $P$, one can obviously construct $F$, but $F$ can also be used to reconstruct $P$:

Given a cdf $F$. The according probability measure $P$ is the unique measure on $(mathbbR, mathcalBmathbbR)$, such that $P((a,b]) = F(b)-F(a)$. One can prove that $P$ exists and that it is unique, using the Carathéodory Extension Theorem.

share|cite|improve this answer

answered Aug 2 at 18:44

Jonas

259210

add a comment | 

up vote
2
down vote

The concept of a cumulative distribution functions (cdf) is closely related to Carathéodory Extension Theorem. The construction is fairly similar to the Lebesgue measure though.

Let $P$ be a probability measure on $(mathbbR, mathcalBmathbbR)$. The cdf of $P$ is the function $$F: mathbbR rightarrow [0,1], x mapsto P((-infty, x]).$$ Given the measure $P$, one can obviously construct $F$, but $F$ can also be used to reconstruct $P$:

Given a cdf $F$. The according probability measure $P$ is the unique measure on $(mathbbR, mathcalBmathbbR)$, such that $P((a,b]) = F(b)-F(a)$. One can prove that $P$ exists and that it is unique, using the Carathéodory Extension Theorem.

share|cite|improve this answer

answered Aug 2 at 18:44

Jonas

259210

add a comment | 

up vote
2
down vote

up vote
2
down vote

The concept of a cumulative distribution functions (cdf) is closely related to Carathéodory Extension Theorem. The construction is fairly similar to the Lebesgue measure though.

Let $P$ be a probability measure on $(mathbbR, mathcalBmathbbR)$. The cdf of $P$ is the function $$F: mathbbR rightarrow [0,1], x mapsto P((-infty, x]).$$ Given the measure $P$, one can obviously construct $F$, but $F$ can also be used to reconstruct $P$:

Given a cdf $F$. The according probability measure $P$ is the unique measure on $(mathbbR, mathcalBmathbbR)$, such that $P((a,b]) = F(b)-F(a)$. One can prove that $P$ exists and that it is unique, using the Carathéodory Extension Theorem.

share|cite|improve this answer

answered Aug 2 at 18:44

Jonas

259210

The concept of a cumulative distribution functions (cdf) is closely related to Carathéodory Extension Theorem. The construction is fairly similar to the Lebesgue measure though.

Let $P$ be a probability measure on $(mathbbR, mathcalBmathbbR)$. The cdf of $P$ is the function $$F: mathbbR rightarrow [0,1], x mapsto P((-infty, x]).$$ Given the measure $P$, one can obviously construct $F$, but $F$ can also be used to reconstruct $P$:

Given a cdf $F$. The according probability measure $P$ is the unique measure on $(mathbbR, mathcalBmathbbR)$, such that $P((a,b]) = F(b)-F(a)$. One can prove that $P$ exists and that it is unique, using the Carathéodory Extension Theorem.

share|cite|improve this answer

answered Aug 2 at 18:44

Jonas

259210

share|cite|improve this answer

share|cite|improve this answer

answered Aug 2 at 18:44

Jonas

259210

answered Aug 2 at 18:44

Jonas

259210

answered Aug 2 at 18:44

Jonas

259210

259210

add a comment | 

add a comment | 

 

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