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Comment by eachro | Hacker News Reader

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 measure theory after numerous gos at it but I haven't ever been able to piece it all together to have a cohesive understanding of the area the way I do for say linear algebra.

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> can you say a bit about how it influences your every day thinking?

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.

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.

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.

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...

Intuitively for me probability theory is a bit clunky before measure theory. Like we have a different equation for expectations if something is continuous vs a discrete distribution. We use probability density functions (pdf) vs a probability mass function (pmf) and etc.

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.

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