# Integration in Hilbert Space

• A. V. Skorohod
Book

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 79)

1. Front Matter
Pages I-XII
2. A. V. Skorohod
Pages 1-26
3. A. V. Skorohod
Pages 27-56
4. A. V. Skorohod
Pages 57-101
5. A. V. Skorohod
Pages 102-139
6. A. V. Skorohod
Pages 140-168
7. Back Matter
Pages 169-180

### Introduction

Integration in function spaces arose in probability theory when a gen­ eral theory of random processes was constructed. Here credit is cer­ tainly due to N. Wiener, who constructed a measure in function space, integrals-with respect to which express the mean value of functionals of Brownian motion trajectories. Brownian trajectories had previously been considered as merely physical (rather than mathematical) phe­ nomena. A. N. Kolmogorov generalized Wiener's construction to allow one to establish the existence of a measure corresponding to an arbitrary random process. These investigations were the beginning of the development of the theory of stochastic processes. A considerable part of this theory involves the solution of problems in the theory of measures on function spaces in the specific language of stochastic pro­ cesses. For example, finding the properties of sample functions is connected with the problem of the existence of a measure on some space; certain problems in statistics reduce to the calculation of the density of one measure w. r. t. another one, and the study of transformations of random processes leads to the study of transformations of function spaces with measure. One must note that the language of probability theory tends to obscure the results obtained in these areas for mathematicians working in other fields. Another dir,ection leading to the study of integrals in function space is the theory and application of differential equations. A. N.

### Keywords

Calculation Hilbert space Hilbertscher Raum Integralrechnung Integration Spaces differential equation equation function function space measure

#### Authors and affiliations

• A. V. Skorohod
• 1
1. 1.Institute of Mathematics AN USSRKievRussia

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-65632-3
• Copyright Information Springer-Verlag Berlin Heidelberg 1974
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-65634-7
• Online ISBN 978-3-642-65632-3
• Series Print ISSN 0071-1136