Arlen Brown

The L p spaces

So far in this book we have considered only pointwise, or almost everywhere, convergence of sequences (and nets) of measurable functions. In many problems where Lebesgue integration occurs, several different kinds of convergence appear naturally. In most cases, different kinds of convergence are associated with different vector spaces of (equivalence classes of) measurable functions, and it is the purpose of this chapter to describe some of these spaces and discuss the corresponding types of convergence. One of the most basic requirements for a mode of convergence is that the usual operations of addition and multiplication by scalars preserve convergence. Because of this, we will be dealing with vector spaces endowed with a topology that is compatible with the linear structure. We start therefore with a brief discussion of the concept of a topological vector space . The discussion is carried out for complex vector spaces, but the analogous concepts make sense for real vector spaces as well.

Perturbations by nilpotent operators on Hilbert space

If T is any noncompact, bounded, linear operator on a separable Hubert space H, then there exists a nilpotent (bounded, linear) operator N on H such that N+Tis invertible.

Measure and Integration

This book covers the material of a one year course in real analysis. It includes an original axiomatic approach to Lebesgue integration which the authors have found to be effective in the classroom. Each chapter contains numerous examples and an extensive problem set which expands considerably the breadth of the material covered in the text. Hints are included for some of the more difficult problems

Standard measure spaces

There are many examples of measure spaces that present various pathologies and on which some of the deeper theorems of measure theory fail. Elements of the class of standard measure spaces, to be defined shortly, do not display any of these pathologies and, in addition, can be classified up to a natural notion of isomorphism. The results we present here use little in addition to the observation that a separable metric space can be written as a countable union of closed subsets of arbitrarily small diameter (for instance, the closed balls of a fixed radius centered at the points of a countable dense set).

Convergence theorems for Lebesgue integrals

Lebesgue integration is a powerful tool principally on account of several convergence theorems (Theorems 4.24, 4.29, 4.31, and 4.35), and these are the main focus of this chapter. There are, however, several other things to be established. We begin by introducing the signed and \(\varphi\) defined on a ring always satisfy \(\varphi (\varnothing ) = 0\).

Rings of sets

It is a familiar fact of elementary calculus that the integral of a function exists only if the function is continuous, or nearly so. In the theory of the Lebesgue integral, with which we are concerned in this book, continuity is replaced by a significantly less stringent requirement known as measurability. This concept, in turn, is defined in terms of a certain type of collection of sets, called a \( \sigma \)-algebra, and so we begin with a brief look at this and some related concepts.

Integrals and measures

In the language of modern integration theory the term integral refers to a number of somewhat different concepts, arrived at through a variety of constructions and definitions. About the only thing that can be said about integration in reasonable generality is that an integral on a space X is a linear transformation that is defined on a vector space of functions on X and satisfies certain continuity requirements. As regards the Lebesgue integral, however, matters are in a much less chaotic state. Indeed, while a considerable number of different definitions and constructions can be found in the literature, there is unanimous agreement on what a Lebesgue integral is. We provide an axiomatic characterization.

Measure and topology

Given a topological space X, there is a natural \(\sigma\)-algebra of subsets of X, namely the \(\sigma\)-algebra B X of Borel sets. When X is locally compact (that is, every point has a relatively compact neighborhood) another useful \(\sigma\)-algebra is the \(\sigma\)-algebra generated by the compact G δ subsets of X. This collection is called the \(\sigma\)-algebra of Baire sets and is denoted Ba X . The \(\sigma\)-algebra Ba X is defined the same way for arbitrary topological spaces, but it is not so useful when X is not locally compact.

Signed measures, complex measures, and absolute continuity

Given a measure, a signed measure, or a complex measure μ on a measurable space (X, S), there are frequently other measures nearby that are closely related to μ in one way or another. These measures can often be put to good use in explicating various properties of μ. In the same vein, given two measures μ and ν (on the same or different measurable spaces), there are often useful ways in which they can be related and other ways of constructing new measures from the pair. In this chapter we explore some of the most important of these constructions and relations.

Existence and uniqueness of measures

As was shown in Chapter 3, for any measurable space (X, S), the correspondence between the set of measures on (X, S) and the set of Lebesgue integrals on (X, S) is a bijection (Theorems 3.29 and 3.37). This knowledge is of small value, however, unless one has in hand a good supply of measures to be integrated with respect to. In this chapter we discuss some of the more important ways in which measures arise.