Functions and Integrals
Chapter 2 is devoted to the definition and basic properties of the Lebesgue integral. functions, the functions that are simple enough that the integral can be defined for them, if their values are not too large (Section 2.1). After a brief look in Section 2.2 at properties that hold almost everywhere (that is, that may fail on some set of measure zero, as long as they hold everywhere else), we turn to the definition of the Lebesgue integral and to its basic properties (Sections 2.3 and 2.4). The chapter ends with a sketch of how the Lebesgue integral relates to the Riemann integral (Section 2.5) and then with a few more details about measurable functions (Section 2.6).
KeywordMeasurable function Borel function Cantor set Cantor function Almost everywhere Integral Integrable function Lebesgue integral Monotone convergence theorem Fatou’s lemma Dominated convergence theorem Riemann integrability Riemann integral
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