Papers › A Continuous Paradoxical Colouring Rule Using Group Action
A Continuous Paradoxical Colouring Rule Using Group Action
Tugkan Batu, Robert Samuel Simon, Grzegorz Tomkowicz
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Given a probability space (X, B, m), measure preserving transformations g₁, …, gₖ 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 and the colours of the finitely g₁ (x), …, gₖ(x) with gᵢ(x) ≠ x for all i and almost all x. We represent a colouring rule as a correspondence F defined on X×Cᵏ with values in C. A function f: X→C satisfies the rule at x if f(x) ∈F( x, f(g₁ x), …, f(gₖ x)). A colouring rule is paradoxical if it can be satisfied in some way almost everywhere with respect to m, but not in {\bf any} way that is measurable with respect to a finitely additive measure that extends the probability measure m and for which the finitely many transformations g₁, …, gₖ remain measure preserving. We show that a colouring rule can be paradoxical when the g₁, …, gₖ are members of a 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(g₁(x) ), …, c(gₖ(x))) for a non-empty upper-semi-continuous convex-valued correspondence F defined on X×Cᵏ. We show that any colouring that approximates the correspondence by ϵ for small enough positive ϵ cannot be measurable in the same finitely additive way. Furthermore any function satisfying the colouring rule illustrates a paradox through finitely many measure preserving shifts defining injective maps from the whole space to subsets of measure summing up to less than one.
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