L^p-Spaces

Abstract

For (X, ℜ, μ) a positive measure space, it has already been noted that μ - a.e. equality is an equivalence relation, and the relation ≤ μ-a.e. a preorder, on.This section studies the structure of the equivalence classes into which μ-a,e. equality partitions.Since the set X/X( ℜ) is always u-null (2.7.7 a)), only the function values on the set X(ℜ) have any significance when equivalence classes are formed: whether we form equivalence classes by partitioning or by partitioning^X(ℜ) the resulting structures will be isomorphic. Nevertheless, it is natural to allow functions on an arbitrary X ⊃ X(ℜ). Our choice is to form μ-equivalence classes by partitioning the set X(ℜ). For arbitrary X ⊃ X(ℜ), we then associate to f ∈ the μ-equivalence class determined by the restriction of f to X(ℜ). This choice simplifies matters somewhat when we work with different sets at the same time.