It was recently pointed out by Bill Johnson in a comment to a question concerning "Fully exchangeable random sequences" that if an infinite sequence of random variables is exchangeable, then it is "fully exchangeable", that is to say its distribution is in fact invariant by any permutation (even if the permutation has no fixed point). The same argument shows that the distribution of an infinite exchangeable sequence of random variables is invariant under the shift map. Hence you can view (A) as the pointwise ergodic theorem for the shift map.
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Removed the present link of the deleted question and provided the archived one.
Thierry de la Rue
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