General topology -- Metric spaces -- Polish spaces, and compactness -- Semi-continuous functions -- Uniform spaces and topological groups -- Càdlàg functions -- Banach spaces -- Hilbert space -- The Hahn-Banach theorem -- Convex functions -- Subdifferentials and the legendre transform -- Compact convex Polish spaces -- Some fixed point theorems -- Abstract measure theory -- Further measure theory -- Borel measures -- Measures on Euclidean space -- Convergence of measures -- Introduction to Choquet theory -- Optimal transportation -- Wasserstein metrics -- Some examples.