Sets of functions and Lebesgue integration

Abstract

In due course we endow sets with particular properties and on the basis of these assumed properties construct a theory for special kinds of sets such as Hilbert spaces. In the development of this theory it is not necessary to appeal to the precise character of a set: the basic axioms, and the theorems that follow from these axioms, apply equally to sets whose members are numbers or matrices or functions. Before embarking on the task of describing this general framework, however, we first introduce two important examples of sets, or spaces (as they are usually called when endowed with additional properties) of functions: these are the spaces of continuous functions, and the L p spaces of functions whose pth powers are integrable. With these at our disposal it is possible in subsequent chapters to illustrate aspects of the general theory, using as special examples sets such as ℝ or ℝn which were introduced in the last chapter, as well as spaces of functions.KeywordsContinuous FunctionMeasurable FunctionLebesgue MeasureSimple FunctionMeasure ZeroThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.