Statement of the DeFinetti's representation theorem I'm using a standard textbook in statistics [1], and am confused with the statement of the DeFinetti's representation theorem.

Theorem 1.49 (DeFinetti's representation theorem). Let $(S,A,\mu)$ be a probability space, and let $(\mathcal{X},\mathcal{B})$ be a Borel space. For each $n$, let $X_n: S\to \mathcal{X}$ be measurable. The sequence $\{X_n\}_{n=1}^\infty$ is exchangeable if and only if there is a random probability measure $\mathbb{P}$ on $(\mathcal{X},\mathcal{B})$ such that, conditional on $\mathbb{P} = P$, the $X_n$'s are IID with distribution $P$. Furthermore, if the sequence is exchangeable, then the distribution of $\mathbb{P}$ is unique, and $\mathbb{P}_n(B)$ converges to $\mathbb{P}(B)$ almost surely for each $B \in \mathcal{B}$.

More precisely, I'm a confused with what a "random probability measure $\mathbb{P}$ on
$(\mathcal{X},\mathcal{B})$" means. Does it mean a random variable that takes value in
the space $M_{1}(\mathcal{X},\mathcal{B})$ of probability measure on $(\mathcal{X},\mathcal{B})$? This seems to
make the most sense because it later says

... the $X_n$ are IID with distribution P.

Here each $X_{n}$ is a measurable function from $S$ to $\mathcal{X}$, and thus its
distribution is the pushforward measure $X_{n,\star}(\mu)$, which is
an element in $M_{1}(\mathcal{X},\mathcal{B})$.
But if that's the case, then what's the domain of $\mathbb{P}$? I guess the domain is $S$.. it does not say it clearly. If the domain is indeed $S$, can $\mathbb{P}$ be written out in general in terms of the $X_{n}$?
Reference

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*[1] Theory of Statistics-[Mark Schervish]

 A: $\mathbb P$ is a map from $S \times \mathcal B$ to $[0,1]$ such that
a) $E \mapsto \mathbb P (\omega,E)$ is a probability measure on $\mathcal  B$ for each $\omega \in S$
b) $\omega  \mapsto \mathbb P (\omega,E)$ is a measurable function for each $E \in \mathcal B$
Conditioned on $\mathbb P=P$ means conditioned on the event $\{\omega \in S: \mathbb P (\omega,E) =P(E) \forall E \in \mathcal B\}$.
A: Under the hypotheses of the theorem, the sequence $(X_n)_{n=1}^\infty$ converges almost surely. The probability measure $\mathbb P$ is the marginal probability distribution of the random variable $\lim\limits_{n\to\infty} \overline X_n= \lim\limits_{n\to\infty}(X_1+\cdots+X_n)/n.$ The conditional distribution of the whole sequence $(X_n)_{n=1}^\infty$ given the value of $\mathbb P$ is they are i.i.d. and the conditional distribution of each $X_i$ given that value of $\mathbb P$ is that value of $\mathbb P.$
Maybe an example will make things clearer. Suppose $(X_1,X_2,X_3,\ldots)$ is random and takes values in $\{0,1\}^n.$ If the distribution of this whole sequence is exchangeable, then there is some random variable $P$ taking values in $[0,1]$ such that the conditional distribution of $X_1,X_2,X_3,\ldots$ given $P=\text{(some particular number $p\in[0,1]$)}$ is expressed by saying they are are i.i.d. and each is equal to $1$ or $0$ with respective probabilities $p$ and $1-p.$ In this case the random probability measure is a measure on the sigma-algebra of all (four) subsets of $\{0,1\}.$ And $\Pr\left( \lim\limits_{n\to\infty} X_n =P \right) = 1.$
Notice that this means $X_1,X_2,X_3,\ldots$ are positively correlated unless the distribution of $P$ is degenerate, a delta distribution, in which latter case the correlation is zero.
(Exchangeable sequences with negative correlation between terms of the sequence cannot be extended to infinitely long exchangeable sequences.)
