Spaces of integrable functions

Abstract

Until now we have treated the points of the measure space (ℝ, M, m) and, more generally, of any abstract probability space (Ω, F, P),as the basic objects, and regarded measurable or integrable functions as mappings associating real numbers with them. We now alter our point of view a little, by treating an integrable function as a ‘point’ in a function space, or, more precisely, as an element of a normed vector space. For this we need some extra structure on the space of functions we deal with, and we need to come to terms with the fact that the measure and integral cannot distinguish between functions which are almost everywhere equal.