Cannarsa Piermarco - Introduction to Measure Theory and Functional Analysis

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Part I
Measure and Integration

Springer International Publishing Switzerland 2015

Piermarco Cannarsa and Teresa D'Aprile Introduction to Measure Theory and Functional Analysis UNITEXT 10.1007/978-3-319-17019-0_1

1. Measure Spaces

Piermarco Cannarsa 1

(1)

Department of Mathematics, Universit degli Studi di Roma Tor Vergata, Rome, RM, Italy

Piermarco Cannarsa (Corresponding author)

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Teresa DAprile

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The concept of measure of a set originates from the classical notion of volume of an interval in . Starting from such an intuitive idea, by a covering process one can assign to any set a nonnegative number which quantifies its extent. Such an association leads to the introduction of a set function called exterior measure , which is defined for all subsets of . The exterior measure is monotone but fails to be additive. Following Carathodorys construction, it is possible to select a family of sets for which the exterior measure enjoys further properties such as countable additivity. By restricting the exterior measure to such a family one obtains a complete measure . This is the procedure that allows to define the Lebesgue measure in . The family of all Lebesgue measurable sets is very large: sets that fail to be measurable can only be constructed by using the Axiom of Choice.

Although the Lebesgue measure was initially developed in euclidean spaces, this theory is independent of the geometry of the background space and applies to abstract spaces as well. This fact is essential for applications: indeed measure theory has been successfully applied to functional analysis, probability, dynamical systems, and other domains of mathematics.

In this chapter, we will develop measure theory from an abstract viewpoint, extending the procedure that leads to the Lebesgue measure in order to construct a large variety of measures on a generic space . In the particular case of , a special role is played by Radon measures (of which the Lebesgue measure is an example) that have important regularity properties.

1.1 Algebras and -Algebras of Sets

1.1.1 Notation and Preliminaries

We shall denote by a nonempty set, by the set of all parts (i.e., subsets) of , and by the empty set.

For any subset of we shall denote by its complement, i.e.,

For any we set .

Let be a sequence in . The following De Morgan identity holds:

We define

If , then we set , and we say that converges to (in this sense we shall write ).

Remark 1.1

(a)

As is easily checked, (resp., ) consists of those elements of that belong to infinitely many subsets (resp., that belong to all but a finite number of subsets ). Therefore

(b)

It is also immediate to check that if is increasing ( ), then

whereas, if is decreasing (

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