Paradoxical Decompositions and Colouring Rules

A colouring rule is a way to colour the points x of a probability space according to the colours of finitely many measures preserving tranformations of x. The rule is paradoxical if the rule can be satisfied a.e. by some colourings, but by none whose inverse images are measurable with respect to any finitely additive extension for which the transformations remain measure preserving. We demonstrate paradoxical colouring rules defined via u.s.c. convex valued correspondences (if the colours b and c are acceptable by the rule than so are all convex combinations of b and c). This connects measure theoretic paradoxes to problems of optimization and shows that there is a continuous mapping from bounded group-invariant measurable functions to itself that doesn't have a fixed point (but does has a fixed point in non-measurable functions).