I am a physicist with a somewhat limited mathematical background. However, I decided to 'break in' in modern mathematical physics, and that meant acquiring first a modicum of functional analysis and the required measure theory, harmonic analysis, and operator theory that goes with it. If you can afford the time, the classical learning path through, e.g., Kolmogorov & Fomin > Rudin (R&C) > Reed & Simon I (a path that I recommend, by the way), with possible excursions into ODEs and PDEs, probability theory, and modern geometry, is the safe way to go. If you cannot afford the time, read Lieb & Loss. It provides a tremendous jump start into what really matters for a beginning mathematical physicist: a little measure theory, L^p spaces, Sobolev spaces, a bit of Fourier analysis and PDEs, and inequalities — lots of them (integral, Sobolev, variational). The main theorems are stated, most with proofs, and put into use.

I had a mathematical physics teacher in graduate school that once said (somewhat half-jokingly) that all you need to know is the monotone and dominated convergence theorems, the Fubini theorem, the Borel-Cantelli lemma, the Euler-Lagrange equations and how to resolve the identity using plane waves. This book by Lieb & Loss is a testimony to his confession.