additive set function

additive set function

[¦ad·əd·iv ¦set ‚fəŋk·shən]
(mathematics)
A set function with the properties that (1) the union of any two sets in the range of the function is also in this range and (2) the value of the function at a finite union of disjoint sets in the range of the set function is equal to the sum of the values at each set in the union. Also known as finitely additive set function.
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A real valued, bounded additive set function on (S, [Sigma]) is one which assigns a real value to each element of (S, [Sigma]), and assigns the sum of the values to the union of two disjoint sets.
A real valued bounded additive set function is called countably additive if it assigns the countable sum of the values to a countable union of disjoint sets.
A real valued bounded additive set function [Phi] on (S, [Sigma]) is called purely finitely additive (see Yosida and Hewitt 1952) if whenever a countably additive function v satisfies:
It is worth noting that a purely finitely additive set function [Phi] on the field of subsets of the integers (Z, [Sigma]) cannot be represented by a sequence of real numbers in the sense that there exists no sequence of positive real numbers, [Lambda] = {[[Lambda].
infinity]] [approaches] R, defines a purely finitely additive set function on the integers which is not representable by a sequence of real numbers.
As seen above in Example 3, [25] and [26], such a function defines a non-negative, bounded, additive set function denoted [Mathematical Expression Omitted] on the field of subsets of the integers Z, (Z, [Sigma]).
They cover measure spaces, the Lebesgue integral, differentiation and integration, the classical Banach spaces, and extensions of additive set functions to measures.