Abstract
In this chapter, we give a systematic account of measure theory. The exposition does not assume any preliminary knowledge of the subject. In Sect. 1.1, we discuss the properties of algebras and semirings in detail and consider various examples. In Sect. 1.3, we introduce the notion of measure and establish its basic properties. The next two sections are devoted to the extension of a measure by the Carathéodory method and to properties of such an extension. Theorem 1.5.1 on the uniqueness of an extension proved here is repeatedly used in the book.
In Sect. 1.6, we study properties of the Borel hull of a system of sets.
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Notes
- 1.
Émile Borel (1871–1956)—French mathematician.
- 2.
This term is not widely accepted, but we temporarily use it, for lack of a better one, instead of the lengthy expression “a non-negative finitely additive set function”.
- 3.
It is instructive to compare this argument with the proof of Theorem 1.2.3.
- 4.
Henri Léon Lebesgue (1875–1941)—French mathematician.
- 5.
Constantin Carathéodory (1873–1950)—a German mathematician of Greek origin.
- 6.
Mikhail Yakovlevich Suslin (1894–1919)—Russian mathematician.
References
Bogachev, V.I.: Measure Theory, vols. 1, 2. Springer, Berlin (2007). 1.1.3, 1.5.1, 4.8.7
Bourbaki, N.: General Topology. Chapters 5–10. Springer, Berlin (1989). 1.1.3
Natanson, I.P.: Theory of Functions of a Real Variable. Frederick Ungar, New York (1955/1961) 1.4.1, 2.4.3, Chap. 4
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Makarov, B., Podkorytov, A. (2013). Measure. In: Real Analysis: Measures, Integrals and Applications. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5122-7_1
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