Walter Rudin, Real and Complex Analysis (used most often)
Typically the first nine chapters of Rudin are covered.
Measures and abstract integration.
Lebesgue measure, Lp spaces.
Hilbert spaces, Banach spaces.
Fourier series. (Optional: Fourier transforms, Fourier inversion, and Plancherel theorems)
Integration on product spaces, Fubini’s theorem.
Introduction to probability — probabilistic terminology, Borel-Cantelli, strong law of large numbers (1-2), independence (8), central limit theorem (9), conditional expectation (6). This could be interwoven with the syllabus above. The numbers refer to where these ideas fit into the syllabus above.