If you want to understand research papers and advanced topics in machine learning and statistics, you'll need to have some background in measuretheoretic probability. Measuretheoretic probability does away with the distinction between continuous and discrete probability, at least from a theory standpoint, and it is an elegant way of thinking about probability in general. The subject itself feels much more like pure math than applied probability, so don't expect it to have a lot of practical utility. Much like calculus, it is the subjects that employ measuretheoretic probability that have the real practical value.
Recommended Books

Probability and Measure Theory
Robert B. Ash and Catherine A. DoléansDade
 Intext exercises
 Solutions to some exercises
This is our favorite text for the more mathematically inclined. It is very well organized. It has the clearest exposition on conditional probability that we have read, and that is a topic that many struggle with. Most importantly, it has solutions to many of the problems in the book.

A Probability Path
Sidney I. Resnick
 Intext exercises
This is a very gentle introduction to measuretheoretic probability that is suitable for more applied readers. The text is conversational and welldeveloped, and the exercises aren't that hard.