Nonmeasurable set

A subset of a measure space that does not belong to the measure’s domain, so the measure axioms do not assign it a value.

Measure space: A set equipped with a sigma-algebra and a measure defined on that sigma-algebra. A nonmeasurable set is a subset of the underlying set that is absent from this structure’s sigma-algebra.
Carathéodory measurability criterion: A condition stating that a set splits every test set’s outer measure additively. Failure of this equality is the standard test for nonmeasurability relative to an outer measure.
Lebesgue measurable set: A subset of Euclidean space satisfying the Carathéodory criterion for Lebesgue outer measure. It is the complementary status: Lebesgue measure assigns it a well-defined value.
Banach–Tarski paradox: A theorem that a solid ball in three dimensions can be partitioned and reassembled into two congruent copies. Its finite pieces are nonmeasurable, preventing ordinary volume from being preserved across the decomposition.
Sigma-algebra: A family of subsets containing the whole space and closed under complements and countable unions. Membership in the sigma-algebra is the criterion for measurability in a measure space.
Vitali set: A subset of the real line containing one representative from each rational-translation equivalence class in an interval. Its translates are disjoint while countably many cover an interval, contradicting any translation-invariant length assignment.
Borel measure: A measure defined on the Borel sigma-algebra of a topological space. A set may fail to be Borel yet become measurable after the measure’s domain is completed.
Nonmeasurable event: A subset of a probability space’s outcome set that is not in its event sigma-algebra. It shows why probability models specify which subsets count as events rather than assigning probabilities universally.
Measure: A countably additive function assigning nonnegative values to sets in a specified sigma-algebra. A set outside the measure’s domain receives no value from that measure.
Axiom of choice: The set-theoretic principle that every family of nonempty sets has a choice function. A choice function selects the representatives used to construct a Vitali set.

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