A measure theoretic paradox from a continuous colouring rule

Given a probability space (X, B, m) , measure preserving transformations g 1, ⋯ , g k of X, and a colour set C, a colouring rule is a way to colour the space with C such that the colours allowed for a point x are determined by that point’s location in X and the colours of the finitely many g 1(x) , ⋯ , g k(x) (called descendants). We represent a colouring rule as a correspondence F defined on X× C k with values in C. A function f: X→ C satisfies the rule at x if f(x) ∈ F(x, f(g 1x) , ⋯ , f(g kx)) . A colouring rule is paradoxical if it can be satisfied in some way almost everywhere with respect to m, but not in any way that is measurable with respect to a finitely additive measure that extends the probability measure m defined on B and for which the finitely many transformations g 1, ⋯ , g k remain measure preserving. We show that a colouring rule can be paradoxical when the g 1, ⋯ , g k are members of a semi-group G, the probability space X and the colour set C are compact sets, C is convex and finite dimensional, and the colouring rule says if c: X→ C is the colouring function then the colour c(x) must lie (m a.e.) in F(x, c(g 1(x)) , ⋯ , c(g k(x))) for a non-empty upper-semi-continuous convex-valued correspondence F. Furthermore we show that this colouring rule has a stability property—there is a positive ϵ small enough so that if the expected deviation from the rule does not exceed ϵ then the colouring cannot be measurable in the same finitely additive way. As a consequence, there is a two-person Bayesian game with equilibria, but all ϵ -equilibria for small enough ϵ are not measurable according to any finitely additive measure that respects the information structure of the game.