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May 08'24
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[Hewitt-Savage 0-1 law] Let [math](\xi_n)_{n\geq 1}[/math] be iid r.v.'s with values in some measurable space [math](E,\mathcal{E})[/math]. The map [math]\omega\mapsto (\xi_1(\omega),\xi_2(\omega),...)[/math] defines a r.v. without values in [math]E^{\N^\times}[/math]. A measurable map [math]F[/math] defined on [math]E^{\N^\times}[/math] is said to be symmetric if

[[math]] F(x_1,x_2,...)=F(x_{\pi(1)},x_{\pi(2)},...) [[/math]]

for all permutations [math]\pi[/math] of [math]\N^\times[/math] with finite support.

Prove that if [math]F[/math] is a symmetric function on [math]E^{\N^\times}[/math], then [math]F(\xi_1,\xi_2,...)[/math] is a.s. constant. [math]Hint:[/math] Consider [math]\F_n=\sigma(\xi_1,x_2,...,\xi_n)[/math], [math]\mathcal{G}_n=\sigma(\xi_{n+1},\xi_{n+2},...)[/math], [math]Y=F(\xi_1,\xi_2,...)[/math], [math]X=\E[Y\mid \F_n][/math] and [math]Z_n=\E[Y\mid \mathcal{G}_n][/math].