Lebesguemessbar

Lebesguemessbar is a coined term used in some mathematical discussions to denote that a set, function, or relation is measurable with respect to the Lebesgue measure. The word blends the surname Lebesgue, after Henri Lebesgue, with the German word messbar (measurable). It is not a formal designation in standard texts, but it appears in informal explanations, problem sets, and expository writing to emphasize measurability within the Lebesgue framework.

For a subset E of R^n, being Lebesguemessbar means that E belongs to the Lebesgue sigma-algebra, i.e., E is Lebesgue-measurable and hence has a Lebesgue measure μ(E) defined (possibly infinite). This class includes all Borel sets and, more generally, any set that differs from a Borel set by a null set. Classic examples include open or closed sets and countable unions; non-measurable examples arise from Vitali-type constructions, illustrating the boundary of Lebesgue measurability.

Usage of the term is primarily pedagogical. It signals a focus on Lebesgue measurability when discussing integration, convergence, and measure-theoretic properties. Because it is not a standard term, precise interpretation relies on explicit reference to Lebesgue measurability in formal literature; readers should treat lebesguemessbar as informal shorthand rather than a rigorous technical label.