Completed notes of the course

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Real and Functional Analysis ∗ Angelo Pasquale, Michele Bucelli † September 15, 2019 ∗Master of Science in Mathematical Engineering at Politecnico di Milano. Unofficial notes from the lectures of Professor Maurizio Grasselli of the Mathematics Department. †With special thanks to Andrea Di Primio, Lorenzo Fiorello and Matteo Contini. 2 Contents Contents 1 1 Elements of set theory 5 1.1 Binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Equivalence classes, quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Finite and infinite sets, cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 About the axiom of choice, Part One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 About the axiom of choice, Part Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Measure theory and Real Analysis 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 σ−algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Topological spaces and σ−algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.1 Properties of measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.2 R∗ and B(R∗) . . . . .…