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Starting this week, I will be teaching an introductory graduate course (Math 275A) on probability theory here at UCLA. While I find myself using probabilistic methods routinely nowadays in my research (for instance, the probabilistic concept of Shannon entropy played a crucial role in my recent paper on the Chowla and Elliott conjectures, and random multiplicative functions similarly played a central role in the paper on the Erdos discrepancy problem), this will actually be the first time I will be teaching a course on probability itself (although I did give a course on random matrix theory some years ago that presumed familiarity with graduate-level probability theory). As such, I will be relying primarily on an existing textbook, in this case Durrett’s Probability: Theory and Examples. I still need to prepare lecture notes, though, and so I thought I would continue my practice of putting my notes online, although in this particular case they will be less detailed or complete than with other courses, as they will mostly be focusing on those topics that are not already comprehensively covered in the text of Durrett. Below the fold are my first such set of notes, concerning the classical measure-theoretic foundations of probability. (I wrote on these foundations also in this previous blog post, but in that post I already assumed that the reader was familiar with measure theory and basic probability, whereas in this course not every student will have a strong background in these areas.)

Note: as this set of notes is primarily concerned with foundational issues, it will contain a large number of pedantic (and nearly trivial) formalities and philosophical points. We dwell on these technicalities in this set of notes primarily so that they are out of the way in later notes, when we work with the actual mathematics of probability, rather than on the supporting foundations of that mathematics. In particular, the excessively formal and philosophical language in this set of notes will not be replicated in later notes.

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