[huac]

Measure theory being necessary to statistics is rather contentious; a better discussion is on Andrew Gelman's blog [1].

My school's PhD stats program does require real analysis before the prelims, but for most intents and purposes, 'multi' and 'linal' (as the cool kids say) should be sufficient for machine learning from a comp sci perspective.

I haven't fully worked through ESLR (Hastie and Tibsharini's advanced version of ISLR posted above) but the majority of the math there is linear algebra with some differential equations and calculus thrown in. I've heard Harvard Stat 210 and Berkeley Stat 205A/B cited as good examples of mathematical stat classes - if you're seriously interested maybe take a look at those syllabi.

[1]: http://andrewgelman.com/2008/01/14/what_to_learn_i/