Non-measurable set

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.