What does countable additivity mean?

Definitions for countable additivity
count·able ad·di·tiv·i·ty

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Wiktionary

  1. countable additivitynoun

    The following property of a measure: that the measure of a countable union of disjoint sets is equal to the sum of their measures.

Wikipedia

  1. countable additivity

    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, μ ( A ∪ B ) = μ ( A ) + μ ( B ) . {\textstyle \mu (A\cup B)=\mu (A)+\mu (B).} 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, μ ( ⋃ n = 1 ∞ A n ) = ∑ n = 1 ∞ μ ( A n ) . {\textstyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n}).} 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 modular set function is equivalent to additive set function; see modularity below.

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Numerology

  1. Chaldean Numerology

    The numerical value of countable additivity in Chaldean Numerology is: 9

  2. Pythagorean Numerology

    The numerical value of countable additivity in Pythagorean Numerology is: 9

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"countable additivity." Definitions.net. STANDS4 LLC, 2024. Web. 29 Feb. 2024. <https://www.definitions.net/definition/countable+additivity>.

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