Section 3 Introduction to probability 

3.1

What is Volume? Introduction to measures

Since the most natural framework to think about probability is the measure theoretical framework, and since measure theory is a very basic topic of common-sense mathematics, a very elementary introduction to the rudimentary part of this theory is given here.

 Basically, the question you must think carefully is: what is the area of a 2D flat figure?

 An authoritative reference may be B Simon,  Real Analysis (AMS 2016) Part 1 Chapter 4,which may not be easily accessible by average (non-mathematical) physicists. See MeasureIntro.pdf .

Thus,  Probability = normalized measure

 ``Probability theory is measure theory with a soul.’’ (M Kac)

3.2

Set theory 

Probably, it is worth learning axiomatic set theory (not only the so-called naive set theory), which was initiated by E Zermelo. I recommend the following book for a solid introduction:

 H. D. Ebbinghaus, J. Flum and W. Thomas: Mathematical Logic

 (UndergraduateTexts in Mathematics, Springer 1984)

The following three books may be of interest.

 The first one is a `graphic novel’ of the history of mathematical logic of the era covered by the second basic reference book. B Russel is the protagonist. The third book is a biography of the author of the second book. J Heijenoort was a bodyguard of Trotsky, was one of the lovers of Frida Kahlo, and the main editor of Collected Works of G\”{o}del.

Lebesgue integral

Since we know roughly what measures are, it is an excellent occasion to learn the theoretical minimum of Lebesgue integral. See the following summary for physicists.

https://www.dropbox.com/s/69ctmfiw3dxvl59/AMII-19%20IntegratonRevisited.pdf?dl=0

3.4

R T Cox derived the basic additive property of probability from an axiomatic characterization of reasonable expectation

The axioms amy be summarized as follows: