Example
Consider the unit circle S, and the action on S by a group G consisting of all rational rotations. Namely, these are rotations by angles which are rational multiples of π. Here G is countable (more specifically, G is isomorphic to ) while S is uncountable. Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset with the property that all of its translates by G are disjoint from X and from each other. The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent (by rational rotations). The set X will be non-measurable for any rotation-invariant countably additive probability measure on S: if X has zero measure, countable additivity would imply that the whole circle has zero measure. If X has positive measure, countable additivity would show that the circle has infinite measure.
Read more about this topic: Non-measurable Set
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