[ryankennedyio]

It will depend a lot on your field, but a solid grasp of fundamental probability theory should be applicable everywhere.

I think this is an excellent overview [1]. Learning probability from a measure theory angle is more difficult to grok compared to the frequentist approach everyone is more familiar with, but I found it much more enjoyable. (I learnt the usual way from doing computer science undergrad, but now re-doing it more rigorously for masters in financial engineering)

[1]: http://www.math.uah.edu/stat/index.html

[tnecniv]

What can you do with the measure theoretic foundations vs the traditional approach? I know graduate classes typically take the measure theoretic approach, but it's never been explained to me why.

[ryankennedyio]

To be honest after one semester, I can't eloquently state it without rambling for a few hours, so I'll direct to this excellent HN comment, which does a much better job than I could:

https://news.ycombinator.com/item?id=14286604