= Finite-prefix invariance of tail events
On the full <Cartesian product> $\Omega=\mathbb R^{\mathbb N}$ with its <product sigma-algebra>, take the coordinate <random variables> $X_j(x)=x_j$. A product-measurable <set> $A$ is a <tail event> if and only if membership is unchanged whenever two <sequences> differ in finitely many coordinates. The <measurability> condition is essential: invariance alone is not a license to assign a <probability> to an arbitrary <subset>.
For the forward implication, fix $m$. An <event> in $\sigma(X_{m+1},X_{m+2},\ldots)$ cannot distinguish points with the same remaining coordinates. Since a <tail event> belongs to this <sigma-algebra> for every $m$, every finite change preserves membership.
For the converse, freeze the first $m$ coordinates to zero. The insertion <map>
$$
i_m(y_{m+1},y_{m+2},\ldots)
=(\underbrace{0,\ldots,0}_{m\text{ entries}},y_{m+1},y_{m+2},\ldots)
$$
is a <measurable function> for the <product sigma-algebras>. Thus $B_m=i_m^{-1}(A)$ is a <measurable set>. Invariance gives $A=\pi_m^{-1}(B_m)$, where the coordinate projection $\pi_m$ discards the first $m$ entries. Hence $A\in\sigma(X_{m+1},X_{m+2},\ldots)$ for every $m$. The same argument works on $\{0,1\}^{\mathbb N}$. On a general <probability space>, coordinates need not admit arbitrary replacements, so the defining <intersection> of <sigma-algebras> remains the reliable formulation.
Back to article page
