Non-measurable Set

A non-measurable set is a mathematical concept where a set of points cannot be assigned a consistent size or measure, such as length, area, or volume, using standard methods. In the context of real numbers, some sets are so "irregular" that there's no way to define their size in a finite, logical way without leading to contradictions. These sets challenge our intuitive understanding of measurement and are important in measure theory, a branch of mathematics that studies how to assign sizes to different sets, especially when dealing with very complex or "pathological" examples.