Measure theory | Additive functions

Sigma-additive set function

In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of k disjoint sets (where k is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely-additive set function (the terms are equivalent). However, a finitely-additive set function might not have the additivity property for a union of an infinite number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, Additivity and sigma-additivity are particularly important properties of measures. They are abstractions of how intuitive properties of size (length, area, volume) of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity; that is, σ-additivity implies additivity. The term is equivalent to additive set function; see modularity below. (Wikipedia).

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Measure Theory 1.2 : Sigma Algebras and the Borel Sigma Algebra

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From playlist Measure Theory

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(PP 1.2) Measure theory: Sigma-algebras

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From playlist Probability Theory

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(PP 1.S) Measure theory: Summary

A brief summary of the material from this section, emphasizing probability measures.

From playlist Probability Theory

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(PP 1.3) Measure theory: Measures

(0:00) Sigma-algebra generated by a collection. (4:12) Examples of sigma-algebras. (7:20) Definition of a measure. (9:48) Definition of a probability measure. A playlist of the Probability Primer series is available here: http://www.youtube.com/view_play_list?p=17567A1A3F5DB

From playlist Probability Theory

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Calculus - Find the limit of a function using epsilon and delta

This video shows how to use epsilon and delta to prove that the limit of a function is a certain value. This particular video uses a linear function to highlight the process and make it easier to understand. Later videos take care of more complicated functions and using epsilon and delta

From playlist Calculus

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Mean and Variance of Normal Distribution

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From playlist Probability

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From playlist Series

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what is sigma notation and how to we use it

👉 Learn how to find the partial sum of an arithmetic series. A series is the sum of the terms of a sequence. An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of n terms of an arithmetic sequence is given by Sn = n/2 [2a + (n - 1)d], where a is

From playlist Series

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From playlist What is a Manifold?

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Annette Bachmayr

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From playlist DART X

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From playlist Workshop on Additive Combinatorics 2020

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zkSNARKs -- Recent progress and applications to blockchain protocols by Chaya Ganesh

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From playlist Foundational Aspects of Blockchain Technology 2020

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From playlist Model Theory

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Elliptic Curves - Lecture 9b - The (Picard) group law

This video is part of a graduate course on elliptic curves that I taught at UConn in Spring 2021. The course is an introduction to the theory of elliptic curves. More information about the course can be found at the course website: https://alozano.clas.uconn.edu/math5020-elliptic-curves/

From playlist An Introduction to the Arithmetic of Elliptic Curves

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From playlist Mathematics

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From playlist Winter School on Stochastic Analysis and Control of Fluid Flow

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From playlist Winter School on Stochastic Analysis and Control of Fluid Flow

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Ex: Sigma Notation - Summation Involving a Quadratic

This video provides a basic example of how to evaluate a summation given in sigma notation. Site: http://mathispower4u.com

From playlist Series (Algebra)

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Omar Fawzi: "New quantum Rényi divergences and applications"

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From playlist Entropy Inequalities, Quantum Information and Quantum Physics 2021

Related pages

Set function | Topological space | Sigma-additive set function | Lebesgue measure | Vector space | Extended real number line | Submodular set function | Volume | Ba space | Group (mathematics) | Modular form | Field of sets | Valuation (geometry) | Directed set | Sequence | Banach algebra | Length | Monoid | Signed measure | Mathematics | Union (set theory) | Real number | Family of sets | Banach limit | Mathematical induction | Subadditive set function | Limit of a sequence | Area | Measure (mathematics) | Inner regular measure | Power set | Open set