Integration and Probability

verfasst von: Paul Malliavin

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

It is a distinct pleasure to have the opportunity to introduce Professor Malliavin's book to the English-speaking mathematical world. In recent years there has been a noticeable retreat from the level of ab straction at which graduate-level courses in analysis were previously taught in the United States and elsewhere. In contrast to the practices used in the 1950s and 1960s, when great emphasis was placed on the most general context for integration and operator theory, we have recently witnessed an increased emphasis on detailed discussion of integration over Euclidean space and related problems in probability theory, harmonic analysis, and partial differential equations. Professor Malliavin is uniquely qualified to introduce the student to anal ysis with the proper mix of abstract theories and concrete problems. His mathematical career includes many notable contributions to harmonic anal ysis, complex analysis, and related problems in probability theory and par tial differential equations. Rather than developed as a thing-in-itself, the abstract approach serves as a context into which special models can be couched. For example, the general theory of integration is developed at an abstract level, and only then specialized to discuss the Lebesgue measure and integral on the real line. Another important area is the entire theory of probability, where we prefer to have the abstract model in mind, with no other specialization than total unit mass. Generally, we learn to work at an abstract level so that we can specialize when appropriate.

Inhaltsverzeichnis

Frontmatter

I. Measurable Spaces and Integrable Functions

Abstract

In this chapter, we follow an axiomatic method of exposition. The interest of the concepts introduced will not appear until Chapter II. We introduce the notion of a measure space, a space endowed with a family of measurable subsets satisfying the axioms of σ-algebra. This approach parallels that of the theory of topological spaces, where a topological space is a space endowed with a family of open subsets. As we will see in Chapter IV, a peculiarity of the concept of a σ-algebra is that it is adapted to the propositional calculus (Boolean algebra). Since negation is an operation of this calculus, this leads to the axiom that the complement of a measurable set is measurable. The fact that σ-algebras are closed under taking complements is an essential difference between the family of open sets of a topological space and the family of measurable sets of a measure space. In order to be able to take limits of sequences, we impose another axiom: A countable union of measurable sets is measurable.

Paul Malliavin

II. Borel Measures and Radon Measures

Abstract

The preceding chapter dealt with abstract measure theory; given an abstract set X, we rather arbitrarily prescribed the σ-algebra B of its measurable subsets. In this chapter, we work in a space X which is locally compact and can be written as a countable union of compact sets. A natural σ -algebra in this context is the Borel algebra B_X. Alocally finite Borel measure is a measure defined on B_X such that every compact set has finite measure. For X metrizable, we prove Lusin’s theorem: If µ is a locally finite Borel measure and A ∈B_X, then for every ∈> 0 there exist an open set O and a closed set F such that F ⊂ A ⊂ O and µ (O - F) <∈. Thus an arbitrary Borel set can be approximated to within ∈ by both an open and a closed set.

Paul Malliavin

III. Fourier Analysis

Abstract

Fourier analysis can be illustrated by analogies from optics. Given a light beam, the goal of spectral analysis is to determine the monochromatic beams it contains; that is, the beams of the form exp\(\left( {\frac{{2i\pi }}{\lambda }t} \right)\). Once a spectral analysis has been carried out, one can ask whether the analysis is exhaustive: is all the energy of the beam really concentrated in the band of frequencies where the spectral analysis was done? One can also ask whether the beam can be reconstructed from its monochromatic components: can spectral synthesis be performed?

Paul Malliavin

IV. Hilbert Space Methods and Limit Theorems in Probability Theory

Abstract

Before we introduce the notion of probability, it seems advisable to describe the type of mathematical model used to represent a physical system.

Paul Malliavin

V. Gaussian Sobolev Spaces and Stochastic Calculus of Variations

Abstract

In Chapter IV, we began by basing probability theory on the theory of abstract measure spaces of Chapter I. We then studied convergence in distribution by means of the Fourier transform on R^d . Thus both abstract integration theory and classical analysis were necessary to obtain the limit theorems of probability theory. These two sources of Chapter IV derive from the dual nature of distributions. Although a distribution is attached to a very abstract object, a random variable on a probability space, it can also be thought of as given by a Radon measure on R. Borrowing an image from Plato, we might say that distributions have a daemonic nature: they come simultaneously from celestial objects (the abstract theory of measure spaces) and terrestrial objects (analysis on R).

Paul Malliavin

Backmatter

Titel: Integration and Probability
verfasst von: Paul Malliavin
Verlag: Springer New York
Electronic ISBN: 978-1-4612-4202-4
Print ISBN: 978-1-4612-8694-3
DOI: https://doi.org/10.1007/978-1-4612-4202-4

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

I. Measurable Spaces and Integrable Functions

II. Borel Measures and Radon Measures

III. Fourier Analysis

IV. Hilbert Space Methods and Limit Theorems in Probability Theory

V. Gaussian Sobolev Spaces and Stochastic Calculus of Variations

Backmatter