Here is a corresponding introduction I found very useful, for readers with advanced undergraduate / graduate level math knowledge.
And this inspiring textbook I think has high overlap with these topics: https://www.amazon.com/Stochastic-Integration-Differential-E...
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.