A measure theoretic paradox from a continuous colouring rule

Abstract

AbstractGiven a probability space $$(X, {{\mathcal {B}}}, m)$$ ( X , B , m ) , measure preserving transformations $$g_1, \dots , g_k$$ 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 $$g_1 (x), \dots , g_k(x)$$ g 1 ( x ) , ⋯ , g k ( x ) (called descendants). We represent a colouring rule as a correspondence F defined on $$X\times C^k$$ X × C k with values in C. A function $$f: X\rightarrow C$$ f : X → C satisfies the rule at x if $$f(x) \in F( x, f(g_1 x), \dots , f(g_k x))$$ f ( x ) ∈ F ( x , f ( g 1 x ) , ⋯ , f ( g k x ) ) . 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 $${{\mathcal {B}}}$$ B and for which the finitely many transformations $$g_1, \dots , g_k$$ g 1 , ⋯ , g k remain measure preserving. We show that a colouring rule can be paradoxical when the $$g_1, \dots , g_k$$ 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 $$c: X\rightarrow C$$ c : X → C is the colouring function then the colour c(x) must lie (m a.e.) in $$F(x, c(g_1(x) ), \dots , c(g_k(x)))$$ 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 $$\epsilon $$ ϵ small enough so that if the expected deviation from the rule does not exceed $$\epsilon $$ 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 $$\epsilon $$ ϵ -equilibria for small enough $$\epsilon $$ ϵ are not measurable according to any finitely additive measure that respects the information structure of the game.