Here is a link to a PDF doc I wrote a few years back: My fast-moving introduction to Advanced Probability Theory.
I was taught undergraduate probability theory by one of the best. Williams’s book Probability with Martingales is a popular introduction to advanced probability theory, and was the text I used to learn about the formal theoretical framework that holds probability together (I highly recommend going through all the exercises at the back of the book).
But what is the best way to teach probability theory? I remember having a discussion with a PhD colleague on whether it is however really necessary to go through all the sigma-algebra part before getting into the interesting measure theory story.
My view has been that advanced probability theory can be taught in a different way, and I always thought that we should put more emphasis on the intuition behind the measure-theory aspects and on how clever the notation is. As an effort to back up my claims, a few years ago I started to write my own Introduction to Probability.
I got about 20 pages down. Uncompleted later chapters were given titles and show the direction I was heading. If I get interesting feedback on what’s there so far I’ll put in the effort to finish this off.
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