Set, Measure and Probability Theory

This book introduces the basic concepts of set theory, measure theory, the axiomatic theory of probability, random variables and multidimensional random variables, functions of random variables, convergence theorems, laws of large numbers, and fundamental inequalities. The idea is to present a seamless connection between the more abstract advanced set theory, the fundamental concepts from measure theory, and integration, to introduce the axiomatic theory of probability, filling in the gaps from previous books and leading to an interesting, robust and, hopefully, self-contained exposition of the theory.

This book also presents an account of the historical evolution of probability theory as a mathematical discipline. Each chapter presents a short biography of the important scientists who helped develop the subject. Appendices include Fourier transforms in one and two dimensions, important formulas and inequalities and commented bibliography. Many examples, illustrations and graphics help the reader understand the theory.

Marcelo Sampaio de Alencar received his Ph.D. from the University of Waterloo in 1994. He has more than 40 years of engineering experience, and he is an IEEE Senior Member. He has been involved with consulting and project development for many companies, agencies, and universities. He is Visiting Professor at the Department of Communications Engineering (DCO), Federal University of Rio Grande do Norte, and a former Full Professor at the Federal University of Campina Grande, Brazil. He is founder and President of the Institute for Advanced Studies in Communications (Iecom). He received the prestigious Attilio Giarola Medal from the Brazilian Microwave and Optoelectronic Society (SBMO). He published over 550 engineering and scientific papers. He has authored 30 books and written chapters for 12 books. Marcelo S. Alencar is editor of the River Publishers Series in Signal, Image and Speech Processing and is: Editor of the Journal of the Brazilian Telecommunication Society (SBrT); Founder of the Journal of Communications and Information Systems (SBrT); Member of the International Editorial Board of the Journal of Communications Software and Systems (JCOMSS); Member of the Editorial Board of the Journal of Networks (JNW); Co-Editor-in-Chief of the Journal of Communication and Information Systems. He has been a member of the IEEE, SBMO and SBrT Boards. He is a Registered Professional Engineer and recipient of two grants from the IEEE Foundation.

Raphael Tavares de Alencar obtained his Bachelor Degree in Electrical Engineering Degree (Majors in Telecommunications and Power Systems), in 2012, and his Master’s Degree in Electrical Engineering (Telecommunications) in 2014, both from the Federal University of Campina Grande, Brazil and he has a Ph.D. from the University of Grenoble, France. He was involved in an International exchange program at the École Nationale Supérieuse des Systèmes Avancés et Réseaux, ESISAR, Valence, France, from 2010 to 2011. He has worked as a consulting engineer for a project for Alpargatas (Havainas© manufacturer and distributer in the country), and his main activities included development, manufacture, installation and testing of a security system to protect mill operators. The main tasks involved electronic and mechanic designs, image processing, sensors and radiofrequency systems testing. He also worked at LCIS – Laboratoire de Conception et d’Intégration des Systèmes, in the development and manufacture of a chipless RFID reader. He was a researcher in a Project funded by RNP (National Research Network): "[CIA]2 – Building Intelligent Cities: from Environmental Instrumentation to Application Development", developed at the Institute of Advanced Communications Studies (Iecom). His activities included mostly research, design and simulation for optical communications and hybrid optical-wireless networks. He is now with Raydiall Automotive, Voiron, France, a joint venture between ARaymond, a world leader in the automotive fastening industry, and Radiall, a world leader in interconnection solutions.