I may appear to be a probability theorist pursuing a masters degree in probability, but I'm actually just a random guy that somehow managed to get to undergraduate limbo. That is, I've concluded my studies but haven't graduated yet I've also had my share of statistical topics, hence why I recognized the term sufficient statistic right away... and remembered the definition making no sense to me. I like the intuition behind unbiased and consistent estimators way more than sufficiency

An introductory course to Measure Theory can give you some useful tools to work with in statistics though! Both the monotone convergence theorem and Lebesgue's dominated convergence theorem are systematically used to prove some properties of certain stastistical models, those related to time series analysis in particular.

That's a neat application that I'm pretty sure you already knew about, and as you said, it's probably (pun intended) for the best that you remember the meaning and usefulness of the law of great numbers as opposed to remembering the whole theoretical background used to get there

Yeah man, probability theory can get ugly if your instructor's approach to abstraction relies solely on the use of pure theoretical concepts to do (and prove) stuff but avoiding the meaning of said stuff. It can get annoying, since often times you need that meaning in order to apply that theoretical knowledge you were just taught. See for example the strong law of great numbers: any random numerical experiment can be dichotomized as a Bernoulli random variable by simply taking an event and its complement, the law of great numbers allows you to estimate the probability of said event by counting the amount of observations that fit into that event and dividing that amount by n , the total amount of observations.

By the way, those sufficient statistics just made me nostalgic. It's been a while since I last saw anything related to them... not that I ever really understood what they were

Ah it's cool that you don't understand the weak law, I hate it 'cause I can't seem to find a real meaning to the so-called "convergence in probability." The strong law is the kewl one as convergence almost everywhere is much clearer to me.