u/Finance_Aderfos

I am self studying measure theory as an finance undergraduate, in order to pursue an Applied Mathematics/ Mathematical Modelling MSc. In the master’s curriculum the main subjects with measure theory as a prerequisite are Probability Theory, Stochastic DE and Non Linear Functional analysis.
I am making this post to understand in what depth i should read measure theory in order to obtain the knowledge needed to work through proofs in those subjects without having much trouble.
Should i learn how to prove “anything” or any exercise in the textbook for example? What proofs should i memorise and what type of exercises should i know how to solve easily in order to face those subjects? Currently studying from Axler’s book.