http://andrewgelman.com/2017/08/02/seemingly-intuitive-low-m...
I disagree on multivar calc. Statistics often makes use of matrix derivatives. I have found it helpful to know.
The only formal prerequisite for learning measure theory is that you should know series and sequences. For a reference, I'm not so sure, maybe Halmos's book. The important parts are probably:
- Monotone convergence theorem
- Dominated convergence theorem
- The construction of the Lebesgue integral
- Fubini's theorem and Tonelli's theorem
I would probably try not to get bogged down in details of construction of measures (unless you like that) and take the Lebesgue measure (essentially length) as given. Also check out the Radon-Nikodym theorem which states that we can always (ish) work with density functions.
If you want to read that book you need real analysis more specifically measure theory (unless that subject is in probability theory for you). You cannot get into the last few chapters without it. Dirichlet Process are described using measures.
I don't believe you need multivar calc or info theory. Info theory stuff are used but not as often. I believe you're slanted toward researcher phd position. Gini index, entropy, etc... and such are taken as given when needed.