Invariant sigma-algebra of an exchangeable sequence ($\mathcal{S}=\{A:T_{\pi }^{-1}A=A\text{ for every finitely supported permutation }\pi \}$)

On the coordinate product space of a sequence, the invariant sigma-algebra consists of measurable events unchanged by every finite permutation of coordinates. Its definition is pointwise and does not depend on the probability law. An exchangeable law makes each such permutation measure preserving. Events in the tail sigma-algebra are invariant because any finite permutation fixes every sufficiently late coordinate. Conversely, 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 $F=1_A$. 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 $\Vert F-\mathbb{E}[F\mid {\mathcal{G}}_n]{\Vert }_1$ 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.