De Finetti theorem

For an infinite sequence of real exchangeable random variables, conditional on its invariant sigma-algebra of an exchangeable sequence, the coordinates are independent and identically distributed. In particular, for Borel sets $A_0,\dots ,A_p$,Here is the core proof. Let ${\mathcal{S}}_m$ denote invariance under permutations of coordinates $0,\dots ,m$. Symmetrization as conditional expectation identifies the empirical average of each indicator with its conditional expectation on ${\mathcal{S}}_m$. The reverse martingale convergence theorem gives its limit conditional on $\mathcal{S}$. Multiplying the averages sums over all ordered coordinate tuples, including repetitions. Symmetrization of the product itself sums over the distinct tuples. The fraction of repeated tuples is at most $\left(\genfrac{}{}{0ex}{}{p+1}{2}\right)/(m+1)$, so these bounded averages have the same limit. This proves the displayed factorization. Permutation invariance also makes the individual conditional laws identical. A regular conditional distribution on the real line packages that common law as a random probability measure, giving the equivalent mixture-of-iid formulation.