NichtlebesgueSkalarraum

NichtlebesgueS... is a term used in some discussions of measure theory to denote a class of subsets of the real numbers that are not Lebesgue measurable. The Lebesgue measure on the real line assigns a length to a wide range of sets, forming the Lebesgue sigma-algebra, which contains all Borel sets and many limits of such sets. However, there exist subsets of R that do not belong to this sigma-algebra, and thus cannot be assigned a Lebesgue measure in a way that preserves the usual properties of length, translation invariance, and countable additivity. A classic example in this context is a Vitali set, constructed using the axiom of choice, which cannot be Lebesgue measurable.

In this framing, a NichtlebesgueS... is any subset of R that fails to be Lebesgue measurable. Such sets are typically obtained through constructions that select representatives from equivalence classes modulo rational translation. The resulting sets demonstrate that the Lebesgue measure cannot be extended to all subsets of R without violating essential properties like translation invariance or countable additivity. As a consequence, the notion of measurability has a clear boundary: all Lebesgue measurable sets form a strict subset of the power set of R.

The term NichtlebesgueS... is not a standard designation in mainstream texts; it is sometimes used informally to highlight the existence and role of non-measurable sets. For standard terminology, one speaks of non-measurable or non-Lebesgue-measurable sets. See also Lebesgue measure, Vitali set, non-measurable set, and axiom of choice.