Earlier quoted context omitted.
seconded, his grad probability classes were some of my favorites a few years ago (also hi!). 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.
Just out of curiousity, can you say a bit about how it influences your every day thinking? And why is measure theory so essential to really understanding probability? I sort of understand why it's necessary to have the language of measure theory and be able to talk about the measures/probabilities of uncountable sets but don't really understand beyond that. I have the equivalent of an undergrads understanding of meas…
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.