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Axiomatic Method of Measure and Integration (IV). Definition of the Lebesgue Integral and its Fundamental Properties (Yoshifumi Ito “Differential and Integral Calculus II”, Chapters 6, 7, 9)

Yoshifumi Ito

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Abstract

In this paper, we define the Lebesgue integral of the Lebesgue measurable function on Rd, (d ≥ 1). Then we study the method of calculation of the Lebesgue integral. Further we clarify the convergence properties of the Lebesgue integral completely. These facts are the new results.

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In this paper, we define the Lebesgue integral of the Lebesgue measurable function on Rd, (d ≥ 1). Then we study the method of calculation of the Lebesgue integral. Further we clarify the convergence properties of the Lebesgue integral completely. These facts are the new results.

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Available abstract

In this paper, we define the Lebesgue integral of the Lebesgue measurable function on Rd, (d ≥ 1). Then we study the method of calculation of the Lebesgue integral. Further we clarify the convergence properties of the Lebesgue integral completely. These facts are the new results.

Key concepts: Mathematics, Lebesgue–Stieltjes integration, Lebesgue integration, Riemann integral, Lebesgue's number lemma, Daniell integral, Dominated convergence theorem, Axiom

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Axiomatic Method of Measure and Integration (IV). Definition of the Lebesgue Integral and its Fundamental Properties (Yoshifumi Ito “Differential and Integral Calculus II”, Chapters 6, 7, 9) — Research Paper | ScholarLens