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)

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.