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.