Document information
- University
- Politecnico di Milano
- Degree programme
- Mathematical Engineering
- Subject
- Real and functional analysis
- Classification
- Notes · Complete set
- Original format
- Text
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NotesComplete set
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Study material for Real and functional analysis, shared by the Studwiz community and reviewed by moderators.
Real and functional analysisComplete set
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Study material for Real and functional analysis, shared by the Studwiz community and reviewed by moderators.
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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∗) . . . . .…
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