Measure Theoretic Probability Theory

Recently, I just completed my undergraduate Real Analysis course at UC Santa Cruz, and for the course, we were required to write a paper. I chose to write on this fantastic and pleasantly rigorous intersection of Measure Theory and Probability Theory — in particular, continuous sample spaces. The traditional method of treating probability theory is with its own axiomatic system and then the complete buildup of the theory from those axioms. However, with this new junction, we can transfer our knowledge from measures to the idea of a probability measure which encapsulates what we traditionally know and understand as the probability function.

For my paper, I spend quite some time developing the preliminary measure theory from the ground up and then delve into the probability theory aspect at the end. For the section on probability theory in the paper, I would recommend noticing the parallel between the theory described in a previous post. I would call this paper a review of this juncture and extremely basic in its presentation. The file for the paper can be found below.

I never planned to include material from measure theory in this blog, but at the same time measure theory is becoming more and more prominent in our fundamental understandings of continuous spaces that we may use in Theoretical Computer Science such as the probability sample space. Since this material intersected with my focus on probability theory on this blog, I have posted this paper here.

Since I don’t plan to continue this topic in the future, I wanted to share some very effective resources for learning the material. Firstly, to get a really good foundational grasp on measure theory, I recommend checking out The Bright Side of Mathematics’ playlist on this which can be found here. Also to learn more on the probability side of measure theory, Alexandar Grigoriyan’s lecture notes on this subject are quite comprehensive and readable — they can be found here.

Looking ahead, I plan to start writing more on randomized algorithms, so look out for those posts!