= Exchangeable invariant events are tail events modulo null sets
For a real sequence of <exchangeable random variables>, every event $A$ in its <invariant sigma-algebra of an exchangeable sequence> agrees almost surely with an event of its <tail sigma-algebra>. Put $F=\mathbf1_A$. Approximate $F$ in $L^1$ by its <conditional expectation> on the first $n+1$ coordinates. Swap that block with the next $n+1$ coordinates. Invariance of $A$ and the law shows that the conditional expectation on the second block is an equally good approximation. It is measurable with respect to the tail after index $n$. The <L1 contraction of conditional expectation> then bounds $\|F-\mathbb E[F\mid\mathcal G_n]\|_1$ by twice the approximation error. The <reverse martingale convergence theorem> gives $F=\mathbb E[F\mid\bigcap_n\mathcal G_n]$ almost surely. Thresholding the latter at $1/2$ supplies the required tail event. Together with the reverse inclusion, the two sigma-algebras have identical probability completions.
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