求问关于real analysis重要性的问题！

认为申请Top Programs的话最好：Elementary real analysis -> point set topology -> graduate measure theoretic probability + stochastic processes (我的学校的sequence有conditional expectation, martingales, markov chains, branching processes, ergodic theory, brownian motion). Measure theory 可以和 graduate probability theory同时学因为overlap很多。过后如果读到Brownian motion的话, functional analysis会有用(至少要见过Arzela-Ascoli吧，这个ODEs课会学）。学graduate probability的主要目的是可以接触proof techniques比如各种coupling.