[siddboots]

Prior to giving real analysis and measure theory a serious go, I feel as though I was carrying around quite a lot of notation baggage that was essentially opaque to me. A lot of it was simply "received knowledge" and not at all cleanly organized in my mind.

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.

[throwawaySG]

Do you think that real analysis and measure theory helped you get a better grip on the notion of a r.v. than just the simple function from sample space to real line definition? I'm slightly tempted to take or at least try to self-study real analysis and eventually measure theory, but everyone I know (including profs) has told me not to bother if I'm not going to do theoretical stuff.

[nicolapede]

It depends what you mean by "getting a better grip". There are books on scientific topics that do not rely on technical details. When they are great, they are so exactly because, even with this constraint, they manage to clearly convey the elemental notions to a layman ([0] is a great example). It is debatable whether the grip you get in this way is better or not. Certainly it can get deeper, when complemented with the right analytical tools.

[0] - https://www.amazon.co.uk/Relativity-Routledge-Classics-Bertr...