ABSTRACT

The key idea in Riemann integration on Rn is the partitioning of the domain of the integrand f into n-dimensional subintervals. The Riemann integral is then obtained as a limit of Riemann sums, that is, sums of function values times the volumes of the subintervals. In Lebesgue integration, it is the range of f rather than the domain that is partitioned into subintervals (see Figure 10.7). This still produces a partition of the domain of f ; however, the sets in this partition are generally more complicated than subintervals. The Lebesgue integral is constructed by multiplying the measure of these sets by function values, adding the results, and then taking limits. In this chapter we construct the measure and in the next chapter we construct the integral. The precise connection between the Riemann and Lebesgue integrals is made in Section 11.4.