= Symmetrization as conditional expectation
{title2=$\mathbb E[Z\mid\mathcal I]=\frac1{|\Gamma|}\sum_{g\in\Gamma}Z\circ T_g$}
If a finite group $\Gamma$ acts by measure-preserving maps on a <probability space>, let $\mathcal I$ be its invariant <sigma-algebra>. For an integrable <random variable> $Z$, the displayed group average is $\mathcal I$-measurable. For $A\in\mathcal I$, changing variables under each group element gives $\int_A Z\circ T_g\,d\mathbb P=\int_A Z\,d\mathbb P$. The average therefore satisfies the defining integral identity for <conditional expectation>. For permutations of $N$ exchangeable coordinates, averaging a product attached to $d$ distinct indices yields the sum over all ordered distinct $d$-tuples divided by the <falling factorial> $(N)_d=N!/(N-d)!$.
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