Introduction

A vector lattice (Riesz space) is a linear space equipped with a lattice ordering which is ‘compatible’ with the linear structure. An ordering is here called a lattice ordering if finite sets possess greatest lower bounds and smallest upper bounds. The theory of statistical experiments abounds with linear spaces which are vector lattices for their ‘natural orderings’. Without specifying the structures we mention a few examples of such spaces.

The space of (equivalence classes) of bounded variables.

The space of (equivalence classes) of real valued variables.

The space of (bounded) continuous real valued functions on a topological space.

The space of differences of sublinear functional on a linear space.

The M-space M(ℰ) of an experiment ℰ.

The space of bounded additive set functions on a given algebra of sets.

The space of finite measures on a given σ-algebra of sets.

The L-space L(ℰ) of an experiment ℰ.

The industrious reader may define the ‘natural’ structures in these examples and check that they really deserve to be called vector lattices.

In this chapter we will give a short introduction to the theory of vector lattices. The theory is illustrated throughout by various spaces which are important to the theory of statistical experiments.

The nuts and bolts are collected in sections 5.2–5.6. Actually it might suffice at a first reading to study sections 5.2, 5.3, theorem 5.4.1 and example 5.4.2, and then look over the contents of sections 5.5 and 5.6. The reader might then proceed and return to this chapter when the need arises.