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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:


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?

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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.

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@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

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