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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.

Let $T$ swap coordinate blocks $0,\dots ,n$ and $n+1,\dots ,2n+1$, fixing all other coordinates. The symmetric law makes $T$ measure preserving, and invariance of the event $A$ gives $F\circ T=F$ pointwise. Put ${\mathcal{B}}_n=\sigma (X_{n+1},\dots ,X_{2n+1})$.

Conditional expectation transforms covariantly under this block swap:To check it, the left side is ${\mathcal{B}}_n$-measurable, and for $C\in {\mathcal{B}}_n$ the change of variables by $T$ sends its defining integral to the conditional-expectation identity on $T(C)\in {\mathcal{E}}_n$, with $F\circ T=F$. Thus the two versions agree. Measure preservation now givesThis is the first requested estimate.

The L1 contraction of conditional expectation follows from the pointwise bound $|\mathbb{E}[Z\mid \mathcal{H}]|\le \mathbb{E}[|Z|\mid \mathcal{H}]$: take expectations to get $\Vert \mathbb{E}[Z\mid \mathcal{H}]{\Vert }_1\le \Vert Z{\Vert }_1$. Applied to a difference, it makes conditional expectation a contraction between any two integrable inputs.

Let $U_n=\mathbb{E}[F\mid {\mathcal{B}}_n]$. Since ${\mathcal{B}}_n\subseteq {\mathcal{G}}_n$, $U_n$ is ${\mathcal{G}}_n$-measurable. HenceCombining this with the other given approximation bound yieldsThe left side does not depend on $n$, and $\epsilon >0$ was arbitrary. It is therefore zero. For a $\mathcal{G}$-measurable version $Y=\mathbb{E}[F\mid \mathcal{G}]$, the event $C=\{Y>1/2\}$ belongs to $\mathcal{G}$ and satisfiesThis proves that exchangeable invariant events are tail events modulo null sets. It does not assert literal pointwise equality of the two events or sigma-algebras before completing them with null sets.