The purpose of this monograph is to provide a systematic account of the theory of noncommutative integration in semi-finite von Neumann algebras. It is designed to serve as an introductory graduate level text as well as a basic reference for more established mathematicians with interests in the continually expanding areas of noncommutative analysis and probability. Its origins lie in two apparently distinct areas of mathematical analysis: the theory of operator ideals going back to von Neumann and Schatten and the general theory of rearrangement invariant Banach lattices of measurable functions which has its roots in many areas of classical analysis related to the well-known Lp-spaces. A principal aim, therefore, is to present a general theory which contains each of these motivating areas as special cases.

We prove that, for a finite-dimensional real normed space V, every bounded mean zero function f ∈ L_∞([0, 1]; V) can be written in the form f = g ◦ T − g for some g ∈ L_∞([0, 1]; V) and some ergodic invertible measure preserving transformation T of [0, 1]. Our method moreover allows us to choose g, for any given ε > 0, to be such that ∥g∥_∞ ⩽ (S_V + ε)∥f∥_∞, where S_V is the Steinitz constant corresponding to V.