In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures. Carathéodory's work on outer measures found many applications in measure-theoretic set theory (outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory.
Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than intervals in or balls in . One might expect to define a generalized measuring function on that fulfills the following requirements:
Any interval of reals has measure
The measuring function is a non-negative extended real-valued function defined for all subsets of .
Translation invariance: For any set and any real , the sets and have the same measure
Countable additivity: for any sequence of pairwise disjoint subsets of
It turns out that these requirements are incompatible conditions; see non-measurable set. The purpose of constructing an outer measure on all subsets of is to pick out a class of subsets (to be called measurable) in such a way as to satisfy the countable additivity property.
Given a set let denote the collection of all subsets of including the empty set An outer measure on is a set function
such that
for arbitrary subsets of
Note that there is no subtlety about infinite summation in this definition. Since the summands are all assumed to be nonnegative, the sequence of partial sums could only diverge by increasing without bound.
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In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers and A set function generally aims to subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.
In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets R of a given set Ω can be extended to a measure on the σ-ring generated by R, and this extension is unique if the pre-measure is σ-finite. Consequently, any pre-measure on a ring containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers. This is an extremely powerful result of measure theory, and leads, for example, to the Lebesgue measure.
Constantin Carathéodory (Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant contributions to real and complex analysis, the calculus of variations, and measure theory. He also created an axiomatic formulation of thermodynamics. Carathéodory is considered one of the greatest mathematicians of his era and the most renowned Greek mathematician since antiquity.
The course is based on Durrett's text book
Probability: Theory and Examples.
It takes the measure theory approach to probability theory, wherein expectations are simply abstract integrals.
Dans ce cours on définira et étudiera la notion de mesure et d'intégrale contre une mesure dans un cadre général, généralisant ce qui a été fait en Analyse IV dans le cas réel.
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