Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system. 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. 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. 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. 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. Indeed, their existence is a non-trivial consequence of the axiom of choice. 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. This means that countable unions, countable intersections and complements of measurable subsets are measurable. 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. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure.
It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets is equal to the sum of the measures of the 'smaller' subsets. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X ( seeDefinition below).
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