Measure (Mathematics)

In mathematics, a measure is a formal generalization of everyday geometric notions such as length, area and volume, and of related ideas such as magnitude, mass and the probability of an event. Informally, a measure is monotonic, so that if one set is contained in another, its measure cannot exceed the larger set's, and it assigns the empty set a measure of zero, matching the idea that an empty region takes up no space. Measures underlie probability theory and integration theory, and can be extended to allow negative values, as with signed measures; the modern, rigorous theory of measures was developed in the late nineteenth and early twentieth centuries by mathematicians including Borel, Lebesgue and Caratheodory, and the idea has since found use well beyond pure mathematics, including in quantum physics. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/