Measure on a σ-Algebra

Carlos S. Kubrusly
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Carlos S. Kubrusly: Catholic University of Rio de Janeiro, Electrical Engineering Department

Chapter 2 in Essentials of Measure Theory, 2015, pp 23-39 from Springer

Abstract: Abstract A function whose domain is a collection of sets is called a set function. A measure is a nonnegative extended real-valued set function satisfying some further conditions. The domain of a measure is a subcollection of the power set $$\wp (X)$$ of a given set X. It is advisable to require that the empty set $$\varnothing $$ and the whole set X itself belong to the domain, and to assign the minimum (zero) for the value of the function at the empty set. It is also convenient to require additivity in the following sense. Assume that every finite union of sets in the domain is again a set in the domain. This indicates that the domain might be an algebra. Then assume that the value of the function at any finite union of disjoint sets in the domain equals the sum of the values of the function at each set. Actually, this leads to a possible definition of a concept of measure (measures defined on an algebra will be considered in Chapter 8 ). However, such an approach lacks an important feature that is needed to build up a useful theory, namely, countable additivity. That is, it is required that the notion of additivity also holds for countably infinite unions of disjoint sets, and so countably infinite unions of sets are supposed to be in the domain of a measure. This compels the domain to be a $$\sigma$$ -algebra.

Keywords: Countable Additivity; Unconditional Convergence; Pairwise Disjoint Sets; Real-valued Measure; Arbitrary Countable Family (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-22506-7_2

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