Measure theory: Signed measures and Hahn-decomposition, Metric outer measures, Radon-Nikodym derivative, Lebesgue decomposition.
Functional Analysis: Dual spaces, weak topologies, the Banach-Alaoglu theorem, adjoint operators, compact operators and their spectrum, Fredholm alternative, Hilbert spaces and operators on Hilbert spaces, the spectral theorem for self-adjoint operators on Hilbert spaces, Fredholm determinants, unbounded operators.
Applications can be chosen among: Fourier analysis, ergodic theory, probability theory, Sobolev spaces, differential equations, geometric measure theory (Hausdorff and other measures).