There is a set of more recent lecture notes here, https://web.ma.utexas.edu/users/gordanz/lecture_notes_page.h..., under the "Introduction to Stochastic Processes" section, FYI.
Measure theoretic probability significantly influenced how I think about nearly everything today, which is as strong an endorsement of these notes as I can muster.
yes, I'd like to hear about that, too. I took Theory of Probability classes, and I appreciate that some complicated stuff is necessary to avoid some neat paradoxes, but must admit that measure theory hasn't taken my thinking or intuition forward at all.
For example, I remember fumbling over a modelling problem involving mixed random variables (that is, random variables with both continuous and discrete parts), and in retrospect the problem was that I just didn't have a clear understanding of what a random variable is, and how it relates to mathematical objects and concepts that I was more familiar with, like functions and vector spaces.
The point, for me, was not about needing to use the language of sigma-algebras to solve the types of problems that I come across in my job (electrical engineering and data analysis). It was more about going through the exercise of constructing the tools that I was using day-to-day, so that I could manipulate them with more confidence and creativity.
[0] - https://www.amazon.co.uk/Relativity-Routledge-Classics-Bertr...
We know this stuff is basically getting at the same underlying quantities. Now imagine a distribution over both continuous and discrete. For example something that measures temperature but breaks after a certain threshold. What does the expectation be for such an instrument? Imagine a distribution on different sized arrays of real numbers. How do you define a valid density function? Measure theory gives you the formalisms for those kinds of problems. You in practice don't need it very often but it keeps you on firm ground when you do.
But I didn't retain much since it wasn't good for building intuition (informal proofs were better for that) and a lot of the corner cases it fixed didn't matter for the real world.
The language of measure theory makes a lot of proofs much shorter and easier to remember though. For example, markov's inequality: https://en.wikipedia.org/wiki/Markov%27s_inequality#In_the_l...