1. Foundations#

In this section, we’re going to breeze over concepts and ideas that, ideally, you’ve been exposed to before giving this book a go. In general, this section is going to just borrow results from your real analysis class, which means that a lot of this stuff is going to be left largely unproven (we’re going to take these results to be true within the scope of this book). Further, a lot of this material is going to be presented with little or no context, and it’s going to be basically entirely definition, theorem, and lemma-driven.

If you aren’t familiar with the concepts that are left unproven, stop immediately! While it’s great to be ambitious, trying to work up a more advanced treatment of mathematics like probability theory without having the requisite background in real analysis is like trying to be a doctor without taking a science class. The ambition is meritous, but it might be too big a slice of pie to chew off.

For some other concepts, I think that the idea should be attainable to you with a real analysis class, but the particular result we’ll use might not be a core concept touched on by every class. For these results, I’ll give a fairly rigorous proof. You can use these proofs to gauge whether you are at a level of mathematical maturity required for this book; if the statements all make sense, and you are able to follow the logic of the proofs as-written, I think you are ready to learn probability theory.

We’ll have the following sections:

  1. Limits, which provides some basic definitions regarding limits, which are a basic concept regarding what happens as sequences proceed off to infinity.

  2. Useful Results, which covers many useful theorems and lemmas that will come up again and again in this book.

  3. Series, which covers many applicable properties of series such as convergence tests, and will be invaluable when we try to make proofs about sequences or series of random variables.

Remark 1.1 (Disclaimer)

There are too many concepts from real analysis to exhaustively list out every single concept you should remember, so this section should not be treated as an exhaustive list where, if you know all of these concepts, you will know all of the real analysis concepts needed for the book.

This section should be treated as a primer where, in my opinion, if you can recall most of these concepts and follow the proofs for the ones you don’t, I think you are ready to tackle the book. You might still find that you have to page back to your real analysis book at times to appreciate some of the future proofs you come across in all their finest detail.