Measures and Probabilities pp 3-25 | Cite as

# Riesz Spaces

## Summary

1.1 This first section introduces the notion of ordered groups and lattices. It might be useful to keep in mind some well known examples, such as **Z** or **R** or the space *C*([a, b]) of real-valued continuous functions on [*a, b*] with their natural orderings.

1.2 It is devoted to a short study of Riesz spaces, that is, ordered vector spaces which are lattices. The space *X* of real-valued continuous functions on [*a, b*], endowed with its usual ordering, is an example of Riesz space.

1.3 In this section we turn our attention to the duals of Riesz spaces. If *E* is any ordered vector space, there is a natural notion of positive linear form: a linear form *I* is said to be positive if the image of any positive element of *E* under *I* is a positive real number. Notice that if *E* is the space of real-valued continuous functions on a closed interval [*a, b*], equipped with its natural ordering, a positive linear form on *E* is continuous.

1.4 We finally define Daniell measures. A Daniell measure is a linear form *L* on a space of functions H which has a “finite variation” (Definition 1.4.1) and such that *L*(*f*_{ n }) → 0 whenever *f*_{ n } decreases to 0 in H. This last condition should be understood as a continuity condition. Among the examples that the reader may want to keep in mind axe the Riemann integral considered as a positive linear form on the set of continuous functions on [*a, b*] and also the positive linear forms ε_{ c }: *X* ∋ *f* → *f*(*c*) where *c* ∈ [*a, b*].

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