Suppose one should optimize at a point x in a compact probability space  based on what happens at phi_k x for finitely many ergodic operators phi_k.  If the optimization rule is continuous and there exists an optimal solution, does there exist one that is measurable with respect to some finitely additive measure for which the phi_k remain measure preserving?  The answer is no, there may be uncountable many solutions, all of which are so radically non measurable and such that the fibres of every solution define a paradoxical decomposition. The answer is related to the canonical action of a group G on the Cantor set 2^G.