Abstract
In this chapter we shall introduce all basic concepts of measure theory, adopting the point of view of measures as set functions. The domains of measures may have different stability properties, and this leads to the concepts of ring, algebra and σ-algebra. The most basic tool developed in the chapter is Carathéodory’ theorem, which ensures in many cases the existence and the uniqueness of a σ-additive measure having some prescribed values on a set of generators of the σ-algebra. In the final part of the chapter we will apply these abstract tools to the problem of constructing a “length” measure on the real line, the so-called Lebesgue measure, and we will study its main properties.
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© 2011 Scuola Normale Superiore Pisa
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Ambrosio, L., Da Prato, G., Mennucci, A. (2011). Measure spaces. In: Introduction to Measure Theory and Integration. Appunti/Lecture Notes, vol 10. Edizioni della Normale. https://doi.org/10.1007/978-88-7642-386-4_1
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DOI: https://doi.org/10.1007/978-88-7642-386-4_1
Publisher Name: Edizioni della Normale
Print ISBN: 978-88-7642-385-7
Online ISBN: 978-88-7642-386-4
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