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...

space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. This became the mostly undisputed...

specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events...

precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measures. A measure which has...

mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians...

In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity...

mathematics, the support (sometimes topological support or spectrum) of a measure μ{\displaystyle \mu } on a measurable topological space (X,Borel⁡(X)){\displaystyle...

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...

mathematics, computable measure theory is the part of computable analysis that deals with effective versions of measure theory. Jeremy Avigad (2012), "Inverting...

In mathematics, in particular in measure theory, there are different notions of distribution function and it is important to understand the context in...

In mathematics, in particular in measure theory, a content μ {\displaystyle \mu } is a real-valued function defined on a collection of subsets A {\displaystyle...

Probability Measures, ETH Zürich, Birkhäuser Verlag, Basel, ISBN 3-7643-2428-7 Athreya, Krishna B.; Lahiri, Soumendra N. (2006), Measure theory and probability...

{\displaystyle Y} . Entropy can be formally defined in the language of measure theory as follows: Let ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} be a...

variables Function of bounded variation at Encyclopedia of Mathematics Measure theory Rowland, Todd. "Total Variation". MathWorld.. Jordan decomposition at...

In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff...

finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure. A set in a measure space...

A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set...

properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated...

Netherlands Stevens, S.S. On the theory of scales and measurement 1946. Science. 103, 677–80. Douglas Hubbard: "How to Measure Anything", Wiley (2007), p....

In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a...

information theory (a branch of mathematics studying the transmission, processing and storage of information) is related to measure theory (a branch of...

measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory....

graph. In measure theory, length is most often generalized to general sets of R n {\displaystyle \mathbb {R} ^{n}} via the Lebesgue measure. In the one-dimensional...

d-dimensional Hausdorff measures for any d ≥ 0, which is not necessarily an integer. These measures are fundamental in geometric measure theory. They appear naturally...

In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a...

calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and...

Lebesgue measure. In the mathematical theory of probability, we confine our study to a probability measure μ, which satisfies μ(E) = 1. Lebesgue's theory defines...

In measure theory, or at least in the approach to it via the domain theory, a valuation is a map from the class of open sets of a topological space to...

In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all...

In the mathematical discipline of measure theory, the intensity of a measure is the average value the measure assigns to an interval of length one. Let...