Sublattices of M(X) isometric to M[0,1]

Summary

Introduction

Let X,Y always denote infinite Polish spaces and M(X) stands for the Banach space of Radon measures on X with the variation norm. While the isometric type of sublattices of L₁(X,μ) for some μ ε M(X) is completely understood (they are either l_1,L₁[0,1], or l₁⊕ L₁[0,1], see e.g. [S] III Prop. 11.2) the structure of sublattices of M(X) is much more complicated. Interesting examples of such sublattices are the Henkin-measures on the unit sphere of Cⁿ for n > 1 (see [Rn], Chap. IX), Rajchman measures on the unit circle (see [Ke], Chap. IX), invariant mesures of a family of measurable tranformations of X (see [Ph], Chap. X) and, more generally, the invariant measures of a H-sufficient statistic in the sense of Dynkin (see [Dy], [Ma]). The first two examples are actually bands in M(X) and there is a very nice characterization of such bands in terms of the compact subsets of X that they annihilate due to Mokobodski ([Ke], Chap. IX. 1). The last two examples are usually true sublattices of M(X) isometric to M(0,1).

In this note we characterize sublattices L of the latter kind, (i.e. L is isometric to M(0, 1))in terms of the existence of strongly affine projections, the w*-Radon-Nikodymproperty, martingale compactness, a choquet-type integral representation theorem and finally in terms of the embedding of their unit sphere into M(X) (see section 2 for precise statements).