Suppose there are $(\Omega,\mathcal F,\mathbb P)$ and r.v. $\xi_i$(i$\ge$1)
$\xi_i:(\Omega,\mathcal F,\mathbb P)\to(\mathbb R,\mathcal B,\mu)$
$A\in$ the exchangeable $\sigma$-algebra $\mathcal E $
What does $\mathcal E$ mean?
In my textbook ,there is a descriptive definition:the occurrence of $A$ is not affected by rearranging finitely many of $\xi_i$.
I don't quite understand and I want a mathematical description:
Does it mean: if $\omega\in A$,then $(\xi_1,\xi_2,\xi_3)(\omega)=(\xi_2,\xi_1,\xi_3)(\omega)$? (just an example)
but this understanding has nothing to do with $A$,only the property of $\omega\in\Omega$..
In another textbook,the author first define a finite permutation $p$ in $\mathbb N$
then define a permutation $T_p$ in $\mathbb R^\infty$ by $(x_1,x_2,\dots)\to(x_{p(1)},x_{p(2)},\dots)$
$C\in\mathcal B^\infty$ is called a symmetry set if $C$ is $T_p$ invariant for all $p$,the collection of all symmetry sets is called the exchangeable $\sigma$-algebra.
What's the relationship of two exchangeable $\sigma$-algebra? One in $\Omega$, one in $\mathbb R^\infty$.
Best Answer
Defining $\xi=(\xi_1, \xi_2, \ldots)$ we get a $\mathcal{F}$-measurable function with image on $\mathbb{R}^\mathbb{N}$. If we only care about statements involving the values of $\xi$, the $\sigma$-algebra $\mathcal{F}$ may be too fine. It would be sufficient to consider the $\sigma$-algebra generated by the sets $\xi^{-1}(B)$, where $B\in\mathcal{B}^\infty$.
Going further, if we only care about symmetric statements, then we can further reduce this $\sigma$-algebra to those generated by the sets $\xi^{-1}(C)$ where $C\subseteq \mathbb{B}^\infty$ is a symmetry set. Hence we get a $\sigma$-algebra on $\Omega$ from the exchangeable $\sigma$-algebra.