Measure, Probability and Functional Analysis

This textbook offers a self-contained introduction to probability, covering all topics required for further study in stochastic processes and stochastic analysis, as well as some advanced topics at the interface between probability and functional analysis.

The initial chapters provide a rigorous introduction to measure theory, with a special focus on probability spaces. Next, Lebesgue integration theory is developed in full detail covering the main methods and statements, followed by the important limit theorems of probability. Advanced limit theorems, such as the Berry-Esseen Theorem and Stein’s method, are included. The final part of the book explores interactions between probability and functional analysis. It includes an introduction to Banach function spaces, such as Lorentz and Orlicz spaces, and to random variables with values in Banach spaces. The Itô–Nisio Theorem, the Strong Law of Large Numbers in Banach spaces, and the Bochner, Pettis, and Dunford integrals are presented. As an application, Brownian motion is rigorously constructed and investigated using Banach function space methods.

Based on courses taught by the authors, this book can serve as the main text for a graduate-level course on probability, and each chapter contains a collection of exercises. The unique combination of probability and functional analysis, as well as the advanced and original topics included, will also appeal to researchers working in probability and related fields.

- 1. Introduction – with two examples.- 2. Measure spaces and probability spaces.- 3. Construction of measure spaces.- 4. *Metric and Banach spaces.- 5. *Measures on metric spaces.- 6. Random variables and measurable maps.- 7. Independence.- 8. Integration.- 9. Convergence of random variables.- 10. The theorem of Radon-Nikodym and conditional expectation.- 11. Fourier transform and Gaussian distributions.- 12. Weak convergence.- 13. Strong law of large numbers.- 14. An ergodic theorem.- 15. Limit theorems for weak convergence.- 16. Fourier inversion formulas.- 17. Norm estimates for the Fourier transform.- 18. Riesz representation theorems.- 19. Banach function spaces.- 20. Probability in Banach spaces.- 21. Law of iterated logarithm.- 22. An application to non-life insurance.

Hannah Geiss is a Senior Lecturer of Mathematics at the University of Jyväskylä (Finland), where she has been employed since 2001. She obtained her PhD at the Friedrich Schiller University Jena (Germany). From 2010 to 2014, she served as an Assistant and Associate Professor at the University of Innsbruck (Austria). Her research interests lie in stochastic analysis and stochastic processes.

Stefan Geiss is a Professor of Mathematics at the University of Jyväskylä (Finland). He obtained his PhD and habilitation from the Friedrich Schiller University Jena (Germany). After research stays at the University of Paris 6 (as a Marie Curie Fellow) and the Technical University of Vienna (Austria), he accepted a professorship at the University of Jyväskylä in 2000. From 2009 to 2014, he held a professorship at the University of Innsbruck before returning to Jyväskylä. Stefan Geiss’s research focuses on probability and functional analysis, with a particular emphasis on the interconnections between the two fields.