Let $X = (X_n)_{n\in\mathbb{N}}$ be a stochastic process with values in $E$. For $n \in \mathbb{N}$, define $$\mathcal{E}'_n := \sigma\bigl(F : F : E^\mathbb{N} \rightarrow \mathbb{R} \text{ is measurable and $n$-symmetric}\bigr)$$ and let $\mathcal{E}_n := X^{-1}(\mathcal{E}'_n)$ be the $\sigma$-algebra of events that are invariant under all permutations $\rho \in S(n)$. Further, let $$\mathcal{E}' := \bigcup^\infty_{n=1} \mathcal{E}'_n = \sigma\bigl( F : F : E^\mathbb{N} \rightarrow \mathbb{R} \text{ is measurable and symmetric}\bigr)$$ and let $\mathcal{E} := \bigcap^\infty_{n=1} \mathcal{E}_n = X^{-1}(\mathcal{E}')$ be the $\sigma$-algebra of exchangeable events for $X$, or briefly the exchangeable $\sigma$-algebra.
Can somebody give me some hints how to understand all these strange $\sigma$-algebras?
If we just look at $\mathcal{E}'_n$ :
There are so many completely different $n$-symmetric measurable maps $F: E^\mathbb{N} \rightarrow \mathbb{R}$, like, in the case $E=\mathbb{R}$, $F(x_1, \ldots) = \exp(x_1 + \cdots + x_n)$ or $G(x_1, \ldots) = \sin(x_1) \cdot \sin(x_2) \cdots \sin(x_n)$.
If $\mathcal{E}'_n$ is just the $\sigma$-algebra of sets $M\in\mathcal{B}\bigl(E^{\mathbb{N}}\bigr)$ with the property $$\forall \; (x_1, \ldots, x_n, \ldots) \in M \Longrightarrow \bigl(x_{\rho(1)}, \ldots, x_{\rho(n)}, \ldots\bigr) \in M \text{ for any } \rho\in S_n \, ,$$ why this complicated definition?