In set theory and measure theory, a non-measurable set is a subset of a given space (typically, the real numbers) that cannot be assigned a Lebesgue measure in a consistent way. The concept of measurability is crucial in mathematics, particularly in analysis and probability theory, as it allows for the generalization of notions like length, area, and volume. The existence of non-measurable sets is typically demonstrated using the Axiom of Choice.
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A subset of a measurable space is non-measurable when . On the real line, unless another sigma-algebra is specified, this commonly means that is not a Lebesgue measurable set. By completeness of Lebesgue measure, such a set cannot be contained in a Lebesgue null set.
For an example, use the axiom of choice to select one representative from each equivalence class of under if , and call the resulting set . The translates , for , are pairwise disjoint, their union contains , and their union is contained in . If were a Lebesgue measurable set, translation invariance and countable additivity would make that union have measure zero when and infinite measure when . Both conclusions contradict these inclusions. Thus is non-measurable.