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Measurable Functions

In: Essentials of Measure Theory

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  • Carlos S. Kubrusly

    (Catholic University of Rio de Janeiro, Electrical Engineering Department)

Abstract

The power set $$\wp (X)$$ of a given set X is the collection of all subsets of X. We will work with set functions (i.e., functions whose domains are sets of sets) from Chapter 2 onwards. To begin with, we might assume that a natural candidate for the domain of such functions of sets would be the power set $$\wp (X)$$ of a given set X. This indeed would be an admissible candidate. However, as we will see in subsequent chapters, there are instances where the power set is too large a set to be the domain of some set functions we wish to consider. This means that some functions may lose essential properties if their domain is too large or, in other words, some functions would not be well-behaved when defined on a domain that is as big as a power set (this will be discussed in detail in Chapter 8 ). Given an arbitrary set X, a collection of subsets of X (i.e., a subcollection of the power set $$\wp (X)$$ ) that will be appropriate to our purpose is a $$\sigma$$ -algebra.

Suggested Citation

  • Carlos S. Kubrusly, 2015. "Measurable Functions," Springer Books, in: Essentials of Measure Theory, edition 1, chapter 1, pages 3-21, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-22506-7_1
    DOI: 10.1007/978-3-319-22506-7_1
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