Università Roma Tre

Part 1: Lebesgue Integral in R^n. Definition of L^1 functions. Theorems on limit integration (monotone convergence, dominated convergence, Fatou's Lemma). Completeness of L^1 (Riesz-Fischer Theorem). Iterated integrals and Fubini's theorem. Measurable functions and Lebesgue measure. Convolution and regularization.
Part 2: Fourier Transform in L^1. The L^2 Hilbert Space (on bounded domains and on R^n). Fourier Transform in Schwartz Space. Tempered Distributions. The Fourier Transform in L^2. Plancherel's Theorem.
Part 3: Foundations of the Theory of Ordinary Differential Equations. Examples and classes of ordinary differential equations. Local existence and uniqueness theorem (Picard-Lindelof); Lipschitz dependence on initial data. Maximal and global solutions; globality criteria. Gronwall's lemma and comparison theorems. Linear systems (linear structure, Wronskian); nonhomogeneous systems (variations of constants). Linear systems with constant coefficients (exponential solution). Floquet's theorem.

Terence Tao, An Introduction to Measure theory.
Paolo Acquistapace, lecture notes of Analisi Matematica Due.
Frank Jones, Lebesgue integration on Euclidean space.
Christopher P. Grant, Lecture notes on the Theory of ordinary differential equations.