ABSTRACT
Lebesgue Integration Until the early part of the twentieth century, the standard theory of integration was that of Riemann, which was presented in Chapter 8. Even today, scientists and engineers rarely need to look beyond the Riemann integral in their work. Its definition is simple and clearly motivated as a measure of area, it is well suited to formulating physical laws and performing computations (both analytically and numerically), and it articulates the relationship between integration and differentiation through the fundamental theorem of calculus. The drawback of the theory is that, from an analytical point of view, the class of Riemann integrable functions is not wide enough. Consider the following observations:
1. In Example 8.2 we saw that the function defined on R by
f(x) = ½ 1, x ∈ Q 0, x ∈ Qc (11.1)
is not Riemann integrable on any bounded interval, though the function is almost constant (Q being countable and hence of measure 0).
