Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 8

L^p SPACES

L^p spaces are spaces of functions whose pth power is integrable. For functions in a L^p space, we can define norms and metrics and study the convergence of sequences of functions. In this chapter, we introduce the concepts of L^p spaces and some important inequalities for functions in the L^p spaces.

8.1 Basic Concepts and Facts

Definition 8.1 (L^p Space). Let (S, ∑, μ) be a measure space and p (0, ∞]. Then L^p(S, ∑, μ) is defined as

and

Definition 8.2 (L^p Norm). Let p (0, ∞] and f L^p(S, ∑, μ). The norm of f on L^P(S, ∑, μ) is defined as

When p = ∞, the norm is called the infinite norm or the maximum norm.

Definition 8.3 (Metric Space). Let S be a set and d : S × S → [0, ∞] be a function. (S, d) is called a metric space if for all x, y, z S,

(a) (Symmetry) d(x, y) = d