ABSTRACT
We know that Riemann integral is the most standard and useful enough for applications in physics and engineering. Quite many functions we encounter in practice are continuous (piecewise, at least) so that they are integrable in the sense of Riemann procedure. In advanced subjects including functional analysis, however, we encounter a class of highly irregular functions or even pathological functions that are everywhere discontinuous, to which the concept of ordinary Riemann integral no longer applies to. In order to treat such functions, we need to employ another notion of the integral more flexible than that of the Riemann integral. In this chapter, we give a concise description as to what Lebesgue integral is. The Lebesgue integral not only overcome many difficulties inherent in the Riemann integral, but also provides a basic for constructing a specific class of Hilbert spaces, called Lp spaces.
