CHAPTER VIII - SPACES OF p-th POWER INTEGRABLE FUNCTIONS

Abstract

This chapter presents spaces of p-th power integrable functions. The class of measurable functions f(x) whose p-th power |f(x)|p (p > 0) is integrable on the interval (a, b) will be denoted by Lp (a,b). If the interval of integration is known, we use briefly the symbol LP. It is said that the sequence of functions fn of the class Lp (a, b) is mean convergent of order p to the function f if the distance between the functions fn and f tends to zero, or if the sequence {fn} tends to f, in the metric of the space under consideration. By the uniqueness of the limit function, the uniqueness of the function f(x) can be understood as a point in the space LP functions differing only on a set of measure 0 and are considered as the same point of the space.