Editing Non-measurable set (section)

==Example==
Consider <math>S,</math> the set of all points in the unit circle, and the [[Group action (mathematics)|action]] on <math>S</math> by a group <math>G</math> consisting of all rational rotations (rotations by angles which are [[Rational number|rational]] multiples of <math>\pi</math>). Here <math>G</math> is countable (more specifically, <math>G</math> is isomorphic to <math>\Q/\Z</math>) while <math>S</math> is uncountable. Hence <math>S</math> breaks up into uncountably many [[Orbit (group theory)|orbits]] under <math>G</math> (the orbit of <math>s \in S</math> is the countable set <math>\{ s e^{i q \pi} : q \in \Q \}</math>). Using the [[axiom of choice]], we could pick a single point from each orbit, obtaining an uncountable subset <math>X \subset S</math> with the property that all of the rational translates (translated copies of the form <math>e^{i q \pi} X := \{ e^{i q \pi} x : x \in X \}</math> for some rational <math>q</math>)<ref>{{Cite journal|last=Ábrego|first=Bernardo M.|last2=Fernández-Merchant|first2=Silvia|last3=Llano|first3=Bernardo|date=January 2010|title=On the Maximum Number of Translates in a Point Set|journal=[[Discrete & Computational Geometry]]|language=en|volume=43|issue=1|pages=1–20|doi=10.1007/s00454-008-9111-9|issn=0179-5376|doi-access=free}}</ref> of <math>X</math> by <math>G</math> are [[pairwise disjoint]] (meaning, disjoint from <math>X</math> and from each other). The set of those translates [[Partition of a set|partitions]] the circle into a countable collection of disjoint sets, which are all pairwise congruent (by rational rotations). The set <math>X</math> will be non-measurable for any rotation-invariant countably additive probability measure on <math>S</math>: if <math>X</math> has zero measure, countable additivity would imply that the whole circle has zero measure.  If <math>X</math> has positive measure, countable additivity would show that the circle has infinite measure.