De Finetti: equivalence a.s. according to which measure In Zen`s answer at 
What is so cool about de Finetti's representation theorem?
that is concerned with De Finetti's 0 -1 representation theorem, he says that "De Finetti's law of large numbers" states that it holds that for an exchangeable sequence of RVs $\{X_i\}_{i \in \mathbb{N}}$
$$\lim_{n \to \infty }\overline{X}_n=\Theta \quad \quad \text{almost surely}$$ 
where $\Theta$ is the Random Variable that arises out of De Finetti`s representation theorem.
However it is not clear to me for which measure this equivalence holds almost surely. Is the relevant measure the measure $\mu$ such that  $\mu[\Theta]$ is the prior probability measure?
 A: The analysis ought to be taking place in a general probability space $(\Omega, \mathscr{U}, \mathbb{P})$ with a sample space that encompasses both the observable exchangeable sequence $\mathbf{X}:\Omega \rightarrow \mathbb{R}^\infty$ and the parameter $\Theta:\Omega \rightarrow \mathbb{R}$.  The statement of interest is then taken with respect to the probability measure $\mathbb{P}$, so it can be stated more formally by defining an event positing equivalence, and then saying that this event occurs with probability one under the measure $\mathbb{P}$.  Formally, this is:
$$\mathbb{P} ( \mathcal{E}) = 1 \quad \quad \text{where} \quad \quad\mathcal{E} \equiv \left\{ \omega \in \Omega \Bigg| \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^n X_i(\omega) = \Theta(\omega) \right\} \in \mathscr{U}.$$
O'Neill (2009) argues that the best way to form the parameter $\Theta$ in this analysis is to define it as the Banach limit of the sequence $\mathbf{X}$, which means that it exists for all possible real sequences (i.e., it is well-defined) and it corresponds to the Cesàro limit of the sequence when that exists.  That way, the parameter is equal to the Banach limit of the sequence for all $\omega \in \Omega$ (which is stronger than almost sure equivalence), and $\mathcal{E}$ is the event that the Cesàro limit of the observable sequence exists, which occurs with probability one under the condition of exchangeability.
The prior measure $\mu_\Theta$ for the parameter $\Theta$ is formed from the overall measure $\mathbb{P}$ by restricting attention to events relating to the parameter only.  If we take $\mathscr{U}$ to be the class of all Borel sets in $\mathbb{R}^\infty$ (in order to describe events pertaining to the whole observable sequence) then the domain of the measure $\mu_\Theta$ would be the subspace of Borel sets in $\mathbb{R}$ (to describe the parameter).  The event $\mathcal{E}$ would not be in this domain, and so it would not make sense to try to apply the prior measure $\mu_\Theta$ to describe a probability for this event.
