Paradoxical colouring rules

A colouring rule is a way to determine a function from a probability space to a set of colours based on the colours of finitely many measure preserving transformations. It is paradoxical if there is some function that satisfies the rule however there is no way to satisfy the rule that is measurable with respect to any finitely additive measure for which the transformations remain measure preserving. Paradoxical colouring rules come in discrete and continuous forms. There are continuous maps from the space of functions from a probability space to a bounded finite dimensional simplex without a fixed point in finitely additive measurable functions. With discrete paradoxical colouring rules, there is a graph with finite degree and a number n of colours such that the graph can be coloured properly with n colours however every such proper colouring generates a paradoxical decomposition.