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 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?