Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am working my way through Resnick's A Probability Path, and looking at Exercise 2.5 I am a bit stuck on the application of Dynkin here. The question states:

Problem:

Let P be a probability measure on $ \mathcal{B}(\mathbb{R})$ and. For any $B\in \mathcal{B}(\mathbb{R}),\varepsilon>0 $ there exists a finite union of intervals, A, such that $P(A\Delta B) < \epsilon$.

The text offers a starting point to define a collection of sets, $ \mathcal{G}$, such that:

$ \mathcal{G}= \left \{ B\in \mathcal{B}(\mathbb{R}): \forall\epsilon>0,\exists\cup_nA_{\epsilon}: P(A\Delta B)<\epsilon \right \} $

With help from members of this site, I have made the following progress.

Outline of a solution:

My understanding is that the question is that I need to show that there is a set, A, of finite intervals that approximates B in terms of measures.

To begin with, I have outlined that $ \mathcal{G}$ is a $\lambda $-class (contains $\Omega$, closed under compliments, and closed under finite disjoint unions).

Next, I define a pi-class $ \mathcal{J}=\left\{(a,b):a,b\in\mathbb{R},a<b\right\}$ which generates $ \mathcal{B}$.

So at this point I have $ \mathcal{J}\subset \mathcal{G} \subset\mathcal{B}$ and $\lambda(\mathcal{J})\subset\mathcal{G}$. Then $\sigma(\mathcal{J})= \lambda(\mathcal{J})$ by Dynkin's Theorem, correct?

My question is, where do I go from here? The sigma-field generated by $\mathcal{J}$ is now proven to be a subset of $\mathcal{G}$, but is that all that needs to be shown here? Where does the probability measure come in to play?

share|improve this question

1 Answer 1

up vote 1 down vote accepted
  1. To show that finite open intervals lie in $G$, you have to show that you can approximate them by finite intervals up to measure $\epsilon$. Trivially, you can approximate them by themselves perfectly.

  2. To show that $\Omega=\mathbb{R}\in G$, use the fact that $$P(\mathbb{R})=P\Bigg(\bigcup_{n\in\mathbb{N}}[-n,n)\cup[n, n+1)\Bigg)=\sum_{n\in\mathbb{N}}P\Big([-n,n)\cup[n, n+1)\Big)$$ $$=\lim_{m\to\infty}\sum_{n=1}^m P\Big([-n,n)\cup[n, n+1)\Big).$$ To show that $G$ is closed under complements, use the fact that the complement of a finite union of intervals is again a fnite union of intervals. The latter may be infinite, in which case you can further approximate them by finite intervals. This is an exercise in epsilontics. Finally, if $(A_n)$ is a disjoint sequence in $G$, You can find for each $n$ a finite family of intervals $F_n$ such that $A_n\Delta\bigcup F_n$ has probability less than $1/2^{n+1}\epsilon$. You can approximate the probability of $\bigcup_n A_n$ by the probability of $\bigcup_{n=1}^m A_n$ for $m$ large and you can approxiamte the probability of this again by $\bigcup_{n=1}^m \bigcup F_n$. Doing the details is again an exercise in epsilontics.

share|improve this answer
    
@Justin: You can basically approximate every kind of inteval by every kind of interval. $[a,b]=\bigcap_n (a-1/n,b+1/n)$ and $(a,b)=\bigcup_n [a+1/n,b-1/n]$. –  Michael Greinecker May 20 '12 at 10:15

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.