Measure theory

In mathematics, a measure on a set is a systematic way to assign a number, intuitively interpreted as its size, to some subsets of that set, called measurable sets. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the usual length, area, or volume to subsets of a Euclidean spaces, for which this be defined. For instance, the Lebesgue measure of an interval of real numbers is its usual length.
Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see § Definition, below). A measure must further be countably additive: if a 'large' subset can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets that are measurable, then the 'large' subset is measurable, and its measure is the sum (possibly infinite) of the measures of the "smaller" subsets.
In general, if one wants to associate a consistent size to all subsets of a given set, while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice.
Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.

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