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Orders on locally compact abelian groups

An order on is a subset that satisfies the three axioms: ; ; and . We recall from [1] the following property of orders.

Theorem 2.1   Let be a measurable order on . There are a totally ordered set with largest element ; a chain of subgroups of ; and a collection of continuous real-valued homomorphisms on such that:
(i) for each , is an open subgroup of ;
(ii) if .
Let . Then, for every ,
(iii) for every ,
(iv) for every
(v) When is discrete, ; and when is not discrete, has empty interior and is locally null.

When is discrete, Theorem 2.1 can be deduced from the proof of Hahn's Embedding Theorem for orders (see [13, Theorem 16, p.59]). The general case treated in Theorem 2.1 accounts for the measure theoretic aspect of orders. The proof is based on the study of orders of Hewitt and Koshi [18].

For with , let

 (11) (12)

For , set

 (13)

Note that when is discrete, , and so in this case.

If is a subset of a topological space, we will use and to denote the closure, respectively, the interior of .

Remarks 2.2   (a) It is a classical fact that a group can be ordered if and only if it is torsion-free. Also, an order on is any maximal positively linearly independent set. Thus, orders abound in torsion-free abelian groups, as they can be constructed using Zorn's Lemma to obtain a maximal positively linearly independent set. (See [18, Section 2].) However, if we ask for measurable orders, then we are restricted in many ways in the choices of and also the topology on . As shown in [18], any measurable order on has nonempty interior. Thus, for example, while there are infinitely many orders on , only two are Lebesgue measurable: , and . It is also shown in [18, Theorem (3.2)] that any order on an infinite compact torsion-free abelian group is non-Haar measurable. This effectively shows that if contains a Haar-measurable order , and we use the structure theorem for locally compact abelian groups to write as , where contains a compact open subgroup [20, Theorem (24.30)], then either is a positive integer, or is discrete. (See [1].)
(b) The subgroups are characterized as being the principal convex subgroups in and for each , we have

Consequently, we have if . By construction, the sets are open. For , the subgroup has nonempty interior, since it contains , with . Hence for , is open and closed. Consequently, for , is open and closed.

(c) Let be a continuous homomorphism between two ordered groups. We say that is order-preserving if . Consequently, if is continuous and order preserving, then .

For each , let denote the quotient homomorphism . Because is a principal subgroup, we can define an order on by setting . Moreover, the principal convex subgroups in are precisely the images by of the principal convex subgroups of containing . (See [1, Section 2].)

We end this section with a useful property of orders.

Proposition 2.3   Let be a measurable order on . Then is a -set.

Proof.    If is discrete, there is nothing to prove. If is not discrete, the subgroup is open and nonempty. Hence the set is nonempty, with 0 as a limit point. Given an open nonempty neighborhood of 0, let

Then is a nonempty subset of . Moreover, it is easy to see that , and hence is a -set.

Next: Analyticity Up: Decomposition of analytic measures Previous: Introduction
Stephen Montgomery-Smith 2002-10-30