ABSTRACT
This chapter is devoted to an introductory study of the so-called Lebesgue spaces, or, the "Lp spaces". They have the structure of Banach spaces and are fundamental throughout Analysis, particularly for the theory of Ordinary and of Partial Differential Equations, for Calculus of Variations, for Operator Theory, for Numerical Analysis and for Harmonic Analysis. The chapter begins with the class of Lebesgue functional spaces, over a given fixed measure space. The notion of the essential supremum is a measure-theoretic extension of the usual supremum. In order to achieve our desired approximation result, we require a very strong tool of topology known as "the Urysohn Lemma". This result is typically stated and proved in great generality, but, for the sake of clarity and for the simplicity of the exposition, we isolate and prove a simple special case which is actually all we need for our second main result in this chapter.
