Editing Non-measurable set (section)

==Historical constructions==
The first indication that there might be a problem in defining length for an arbitrary set came from [[Vitali set|Vitali's theorem]].<ref>Moore, Gregory H., Zermelo's Axiom of Choice, Springer-Verlag, 1982, pp. 100–101</ref> A more recent combinatorial construction which is similar to the construction by Robin Thomas of a non-Lebesgue measurable set with some additional properties appeared in American Mathematical Monthly.<ref>{{Cite journal|last=Sadhukhan|first=A.|date=December 2022|title=A Combinatorial Proof of the Existence of Dense Subsets in <math>\mathbb{R}</math> without the "Steinhaus" like Property|journal=[[Am. Math. Mon.]]|language=en|volume=130|issue=2|pages=175|doi=10.1080/00029890.2022.2144665|arxiv=2201.03735}}</ref>

One would expect the measure of the union of two disjoint sets to be the sum of the measure of the two sets. A measure with this natural property is called ''finitely additive''. While a finitely additive measure is sufficient for most intuition of area, and is analogous to [[Riemann integration]], it is considered insufficient for [[probability]], because conventional modern treatments of sequences of events or random variables demand [[countable additivity]].

In this respect, the plane is similar to the line;  there is a finitely additive measure, extending Lebesgue measure, which is invariant under all [[isometries]]. For higher [[dimension]]s the picture gets worse. The [[Hausdorff paradox]] and [[Banach–Tarski paradox]] show that a three-dimensional [[ball (mathematics)|ball]] of radius 1 can be dissected into 5 parts which can be reassembled to form two balls of radius 1.