Measure theory

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. 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. Let be a set and a -algebra over A set function from to the extended real number line is called a measure if the following conditions hold: Non-negativity: For all Countable additivity (or -additivity): For all countable collections of pairwise disjoint sets in Σ, If at least one set has finite measure, then the requirement is met automatically due to countable additivity: and therefore If the condition of non-negativity is dropped, and takes on at most one of the values of then is called a signed measure. The pair is called a measurable space, and the members of are called measurable sets. A triple is called a measure space. A probability measure is a measure with total measure one – that is, A probability space is a measure space with a probability measure. For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures.

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