Measure (mathematics)

Generalization of mass, length, area and volume / From Wikipedia, the free encyclopedia

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In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general.

Informally, a measure has the property of being monotone in the sense that if is a subset of the measure of is less than or equal to the measure of Furthermore, the measure of the empty set is required to be 0. A simple example is a volume (how big an object occupies a space) as a measure.

The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others.

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