Probability and Stochastic Processes

Chapter 1Elements of Probability Measure

The axiomatic approach of Kolmogorov is followed by most books on probability theory. This is the approach of choice for most graduate level probability courses. However, the immediate applicability of the theory learned as such is questionable, and many years of study is required to understand and unleash its full power.

On the other hand, the books on applied probability completely disregard this approach, and they go more or less directly into presenting applications, thus leaving gaps in the reader's knowledge. On a cursory glance, this approach appears to be very useful (the presented problems are all very real and most are difficult). However, I question the utility of this approach when confronted with problems that are slightly different from the ones presented in such books.

I believe no present textbook strikes the right balance between these two approaches. This book is an attempt in this direction. I will start with the axiomatic approach and present as much as I feel will be be necessary for a complete understanding of the theory of probability. I will skip proofs which I consider will not bring something new to the development of the student's understanding.

1.1 Probability Spaces

Let Ω be an abstract set containing all possible outcomes or results of a random experiment or phenomenon. This space is sometimes denoted with S and is named the sample space. I call it “an abstract set” because it could contain anything. For ...