A colouring rule is a way to colour the points x of a probability space according to the colours of finitely many measure preserving transformations of x. The rule is paradoxical if the rule can be satisfied a.e. by some colourings, but by none measurable with respect to any finitely additive extension for which the transformations remain measure preserving. We show that there is a paradoxical colouring rule when the rule is continuous and the measure preserving transformations generate a group. This implies that the only fixed points (as functions of a function space) involve extreme non measurability.