Let $X=(X_1,X_2,...,X_n)$ be an exchangeable $n$-dimensional vector of random variables with values in $\mathbb{R}$, let $\mathfrak{G}$ be the $\sigma$-algebra of permutation-invariant (symmetric) events of $\mathbb{R}^n$, and let $g:\mathbb{R}^n \rightarrow \mathbb{R}$ be a measurable function.
The problem now is to calculate
$$\mathbb{E}[g(X_1,X_2,...,X_n) \mid X^{-1}(\mathfrak{G})].$$
I understand that I am searching for a random variable $Z$ which is $X^{-1}(\mathfrak{G})$-measurable and satisfies $$\int\limits_{A} Z \,\mathrm{d}\mathbb{P} = \int\limits_{A} g(X_1,X_2,...,X_n) \,\mathrm{d}\mathbb{P} $$ for all $A \in X^{-1}(\mathfrak{G})$. I got this far:
$A \in X^{-1}(\mathfrak{G})$ is always of the form $\{X \in B\}$ for one $B \in \mathfrak{G}$.
Due to exchangeability, $Z$ has to satisfy following equality:
\begin{align*} \int\limits_{A} Z \,\mathrm{d}\mathbb{P} &= \int\limits_{A} g(X_1,X_2,...,X_n) \,\mathrm{d}\mathbb{P} \\ &= \int\limits_{B} g(x_1,x_2,...,x_n) \,\mathrm{d}\mathbb{P}^{(X_1,X_2,...,X_n)}(x_1,x_2,...,x_n) \\ &=\int\limits_{B} g(x_1,x_2,...,x_n) \,\mathrm{d}\mathbb{P}^{(X_{\pi(1)},X_{\pi(2)},...,X_{\pi(n)})}(x_1,x_2,...,x_n)\\ &= \int\limits_{A} g(X_{\pi(1)},X_{\pi(2)},...,X_{\pi(n)}) \,\mathrm{d}\mathbb{P} \end{align*}
for any permutation $\pi \in S_n$.
Any hints on how to go on?
Hints (both assertions below to be shown!):
For every $\pi$ in $S_n$, $E[g(X_1,X_2,...,X_n) \mid X^{-1}(\mathfrak{G})]=E[g(X_{\pi(1)},X_{\pi(2)},...,X_{\pi(n)}) \mid X^{-1}(\mathfrak{G})].$
For every measurable function $g$, the random variable $S_g=\sum\limits_{\pi\in S_n}g(X_{\pi(1)},X_{\pi(2)},...,X_{\pi(n)})$ is $X^{-1}(\mathfrak{G})$-measurable.