nonLebesgue

The term "nonLebesgue" typically refers to mathematical concepts, primarily in measure theory, that are distinct from or in contrast to the Lebesgue measure. The Lebesgue measure, developed by Henri Lebesgue, is a fundamental concept in modern analysis that assigns a size or "volume" to subsets of Euclidean space. It is a generalization of the notion of length, area, and volume.

A "nonLebesgue" measure could therefore refer to other types of measures that exist in mathematics. These might include measures that are not defined on all subsets of a space (e.g., only on Borel sets), or measures that have different properties than the Lebesgue measure, such as being discrete, singular, or having a different definition of "size."

For example, the counting measure is a nonLebesgue measure. On a set of discrete points, the counting measure assigns a "size" of one to each individual point. This is different from the Lebesgue measure which would assign zero measure to a single point in Euclidean space.

Another context for "nonLebesgue" might be in the study of spaces where the Lebesgue measure is not the most natural or useful way to define size. This can arise in fractal geometry, where measures are often constructed to capture the intricate structure of self-similar sets, or in certain areas of probability theory where measures are defined on abstract spaces. The concept highlights that while the Lebesgue measure is powerful and widely applicable, it is not the only way to quantify the "size" of sets.