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We show the following generalizations of the de Finetti--Hewitt--Savagetheorem: Given an exchangeable sequence of random elements, the sequence isconditionally i.i.d. if and only if each random element admits a regularconditional distribution given the exchangeable $\sigma$-algebra (equivalently,the shift invariant or the tail algebra). We use this result, which holdswithout any regularity or technical conditions, to demonstrate that anyexchangeable sequence of random elements whose common distribution is Radon isconditional iid.

de Finetti's theorem and the existence of regular conditional distributions and strong laws on exchangeable algebras