Sheldon Axler (Professor Emeritus at San Francisco State University) has a textbook called Measure, Integration & Real Analysis which (as far as I know) covers the basics of measure theory and functional analysis, along with a bit of Fourier analysis and probability theory.
There is an Amazon review of Professor Axler's book that says
... These results are of course very important in that they all have applications in mathematical physics. ...
I would like to know where to look in order to find these applications.
My (mostly uneducated) guess is that I should be looking more towards quantum mechanics and/or maybe particle theory (as opposed to, say, classical mechanics or general relativity). So, perhaps I should be looking at texts like Quantum Theory for Mathematicians by Hall, and Modern Quantum Mechanics by Sakurai and Napolitano.
Does anyone have any guidance to offer? Thanks.