The links above are for studying this as a pure mathematician would. If you want to study it that way, you would take most of the core classes in the undergrad curriculum:

Calculus (without proofs) Linear Algebra Real Analysis (proofs of calculus) Measure Theory

There are also higher level courses that are worth taking, because they motivated a lot of this theory. They would be imo, Functional Analysis (real analysis applied to spaces of functions), and Partial Differential Equations.

If you've knocked off some of the undergrad prereqs and feel good about proofs, this could be the right book for you: https://www.amazon.com/Probability-Martingales-Cambridge-Mat.... Another gem of a book.