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.
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