[mhneu]

No no on measure theory. See below on Feller, Bersekas, Gallager. I think it's best to learn discrete prob first and skip the difficult math until you decide to become a mathematician or really need it for e.g. finance work.

One other objection: yes, those are fields in statistics. But statistics as a field is MUCH broader than departments called statistics. A ton of stats is done in engineering, psychology, and economics (econometrics). In fact I'd say the major research in stats for the past few decades has been done outside of "statistics" as a field. But yes, the books by Tukey, Tufte, and Casella are good.

If the OP is asking about statistics and probability needed for machine learning, he wants to focus on engineering statistics, like: estimation, stochastic processes, and filtering and mathematical statistics. The Wald rec is good and complements Feller.

[madhadron]

The OP is reading Spivak and Dedekind and makes no mention of machine learning. He's interested in the deep mathematics.