Dtjytyk

How do I fix the group tension caused by my character stealing and possibly killing without provocation?

How to test the sharpness of a knife?

El Dorado Word Puzzle II: Videogame Edition

Proving an identity involving cross products and coplanar vectors

Unable to disable Microsoft Store in domain environment

Quoting Keynes in a lecture

How can I safely use "Thalidomide" in my novel while respecting the trademark?

Visualizing the difference curve in a 2D plot?

Given this phrasing in the lease, when should I pay my rent?

Why is participating in the European Parliamentary elections used as a threat?

How do you justify more code being written by following clean code practices?

How would a solely written language work mechanically

Anime with legendary swords made from talismans and a man who could change them with a shattered body

What is the meaning of "You've never met a graph you didn't like?"

If Captain Marvel (MCU) were to have a child with a human male, would the child be human or Kree?

Using streams for a null-safe conversion from an array to list

Has the laser at Magurele, Romania reached a tenth of the Sun's power?

How do I prevent inappropriate ads from appearing in my game?

What does "tick" mean in this sentence?

If the only attacker is removed from combat, is a creature still counted as having attacked this turn?

Why can't the Brexit deadlock in the UK parliament be solved with a plurality vote?

Should I warn new/prospective PhD Student that supervisor is terrible?

Sound waves in different octaves

What should be the ideal length of sentences in a blog post for ease of reading?

Do Vitali sets really explain measure-theoretic probability?

What's the intuition behind and some illustrative applications of probability kernels?Do two probability kernels induce the same distributions if they induce the same distributions on rays?Samples from the Dirichlet measureBetween bayesian and measure theoretic approachesProbabilistic implications of the existence of non-measurable setsShow that there's a $A_n downarrow emptyset$ with $lim_{n to infty} mu(A_n) neq 0$Measure theory and Probability Theory on generalizations of topological spacesWhy keeping a disctinction between almost surely equal elements in probability theory?Proof of a change-of-measure formulaWhy is the sample space called a 'space'?

3

$begingroup$

I’ve read several answers about how how crazy sets like Vitali’s set are (one of the reasons) why we need $sigma$-algebra and measure theory for probability spaces. However, Vitali’s set is only eliminated by requiring non-trivial translation-invariant measures. So either (a) Vitali’s set wasn’t a problem in the first place or (b) probability spaces need translation-invariance.

No definition of probability spaces I’ve seen mentions translation invariance, leaving us with (a): Vitali’s set is not a problem for probabilists. Then:

1. Why do people mention Vitali's set while explaining the measure-theoretic formulation of probability?


2. Why do, in fact, we need measure theory for probability?


3. Are there any other paradoxes arising from uncountable sets to worry about that do apply to regular measures and probability?

probability-theory

share|cite|improve this question

edited Mar 12 at 23:12

Yatharth Agarwal

asked Mar 12 at 21:40

Yatharth AgarwalYatharth Agarwal

542418

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Non-measurable set is not the only reason that we define a measure on $sigma$-algebra instead of the whole power set.
  $endgroup$
  – BigbearZzz
  Mar 12 at 21:48
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @BigbearZzz Some of the other reasons I’ve seen are $sigma$-algebra as “information” which is fine. But still, what’s even the relevance of people mentioning the Vitali set if can’t be solved for all probability spaces anyway?
  $endgroup$
  – Yatharth Agarwal
  Mar 12 at 21:56
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Vitali set serves as an example of a Lebesgue non-measurable set, perhaps the first example of why the Lebesgue outer measure is not a measure on the power set of $Bbb R$. You can say that this has more to do with real analysis than probability theory.
  $endgroup$
  – BigbearZzz
  Mar 12 at 22:01
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You should look into the construction of a probability measure via Caratheodory extension theorem. The presence/absence of Vitali set and translation invariant is not relevant at all, but still the theorem doesn't, in general, let you define a probability measure on the whole power set of your space anyway. This is another reason why $sigma$-algebra is preferred.
  $endgroup$
  – BigbearZzz
  Mar 12 at 22:05
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @BigbearZzz That makes a ton of sense: These Vitali set examples are not actually relevant for probability theory; people just tend to mention them because they’re familiar with analysis, or they’re just trying to present an analogy of how things go wrong with uncountable sets! Does that sound fair?
  $endgroup$
  – Yatharth Agarwal
  Mar 12 at 22:16
  

  
  
  
  

 | 
show 2 more comments

3

$begingroup$

I’ve read several answers about how how crazy sets like Vitali’s set are (one of the reasons) why we need $sigma$-algebra and measure theory for probability spaces. However, Vitali’s set is only eliminated by requiring non-trivial translation-invariant measures. So either (a) Vitali’s set wasn’t a problem in the first place or (b) probability spaces need translation-invariance.

No definition of probability spaces I’ve seen mentions translation invariance, leaving us with (a): Vitali’s set is not a problem for probabilists. Then:

1. Why do people mention Vitali's set while explaining the measure-theoretic formulation of probability?


2. Why do, in fact, we need measure theory for probability?


3. Are there any other paradoxes arising from uncountable sets to worry about that do apply to regular measures and probability?

probability-theory

share|cite|improve this question

edited Mar 12 at 23:12

Yatharth Agarwal

asked Mar 12 at 21:40

Yatharth AgarwalYatharth Agarwal

542418

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Non-measurable set is not the only reason that we define a measure on $sigma$-algebra instead of the whole power set.
  $endgroup$
  – BigbearZzz
  Mar 12 at 21:48
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @BigbearZzz Some of the other reasons I’ve seen are $sigma$-algebra as “information” which is fine. But still, what’s even the relevance of people mentioning the Vitali set if can’t be solved for all probability spaces anyway?
  $endgroup$
  – Yatharth Agarwal
  Mar 12 at 21:56
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Vitali set serves as an example of a Lebesgue non-measurable set, perhaps the first example of why the Lebesgue outer measure is not a measure on the power set of $Bbb R$. You can say that this has more to do with real analysis than probability theory.
  $endgroup$
  – BigbearZzz
  Mar 12 at 22:01
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You should look into the construction of a probability measure via Caratheodory extension theorem. The presence/absence of Vitali set and translation invariant is not relevant at all, but still the theorem doesn't, in general, let you define a probability measure on the whole power set of your space anyway. This is another reason why $sigma$-algebra is preferred.
  $endgroup$
  – BigbearZzz
  Mar 12 at 22:05
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @BigbearZzz That makes a ton of sense: These Vitali set examples are not actually relevant for probability theory; people just tend to mention them because they’re familiar with analysis, or they’re just trying to present an analogy of how things go wrong with uncountable sets! Does that sound fair?
  $endgroup$
  – Yatharth Agarwal
  Mar 12 at 22:16
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

I’ve read several answers about how how crazy sets like Vitali’s set are (one of the reasons) why we need $sigma$-algebra and measure theory for probability spaces. However, Vitali’s set is only eliminated by requiring non-trivial translation-invariant measures. So either (a) Vitali’s set wasn’t a problem in the first place or (b) probability spaces need translation-invariance.

No definition of probability spaces I’ve seen mentions translation invariance, leaving us with (a): Vitali’s set is not a problem for probabilists. Then:

1. Why do people mention Vitali's set while explaining the measure-theoretic formulation of probability?


2. Why do, in fact, we need measure theory for probability?


3. Are there any other paradoxes arising from uncountable sets to worry about that do apply to regular measures and probability?

probability-theory

share|cite|improve this question

edited Mar 12 at 23:12

Yatharth Agarwal

asked Mar 12 at 21:40

Yatharth AgarwalYatharth Agarwal

542418

$endgroup$

share|cite|improve this question

edited Mar 12 at 23:12

Yatharth Agarwal

asked Mar 12 at 21:40

Yatharth AgarwalYatharth Agarwal

542418

share|cite|improve this question

edited Mar 12 at 23:12

Yatharth Agarwal

asked Mar 12 at 21:40

Yatharth AgarwalYatharth Agarwal

542418

asked Mar 12 at 21:40

Yatharth AgarwalYatharth Agarwal

542418

asked Mar 12 at 21:40

Yatharth AgarwalYatharth Agarwal

542418