Holder's Inequality in Spaces of Measurable Functions

Abstract

<p>This thesis gives a historical account of the development of the theory of spaces of measurable functions. The study will be mainly centred around Holder's inequality and its many generalizations.</p> <p>We present a formal axiomatization of the theory including a general form of Holder's inequality. Then we consider various families of spaces of measurable functions, namely the Orlicz spaces and the Lorentz spaces, both of which include the familiar LP-species. In each of these special cases, Holder's inequality is interesting in its own right. As a particular example, the space Llog⁺L and its conjugate, the exponential space, are studied in detail as they are examples of both an Orlicz space and a Lorentz space.</p>