Define measures and by their Fourier transforms: , and . Then we have the following decomposition theorem.
(32) |
(33) |
The next result, crucial to our proof of Theorem 5.1, is already known in the case that with the lexicographic order on the dual. This is due to Garling [15], and is a modification of the celebrated inequalities of Burkholder. Our result can be obtained directly from the result in [15] by combining the techniques of [3] with the homomorphism theorem 4.5. However, we shall take a different approach, in effect reproducing Garling's proof in this more general setting.
Now let us show that if we have the result for compact , then we have it for locally compact . Let denote the quotient homomorphism of onto the discrete group (recall that is open), and define a measurable order on to be . By Remarks 2.2 (c), the decomposition of the group that we get by applying Theorem 2.1 to that group, is precisely the image by of the decomposition of the group . Let denote the compact dual group of . Thus if Theorem 5.2 holds for , then applying Theorem 4.5, we see that Theorem 5.2 holds for .
Henceforth, let us suppose that is compact. We will suppose that the Haar measure on is normalized, so that with Haar measure is a probability space.
Since each one of the subgroups , and ( ) is open, it follows that their annihilators in , , and , are compact. Let and denote the normalized Haar measures on and , respectively. We have (for all ), and (for all ), so that .
For each , let denote the -algebra of subsets of of the form , where is a Borel subset of . We have , whenever . It is a simple matter to see that for , the conditional expectation of with respect to is equal to (see [11, Chapter 5, Section 2]).
We may suppose without loss of generality that . Thus the -algebras form a filtration, and the sequence is a martingale difference sequence with respect to this filtration.
In that case, we have the following result of Burkholder [7, Inequality (1.7)], and [8]. If , then there is a positive constant , depending only upon , such that
Next, by convolving with an approximate identity for consisting of trigonometric polynomials, we may assume that is a trigonometric polynomial. Then we see that for each that the function , , is in . To verify this, it is sufficient to consider the case when is a character in . Then
Now we have the following generalization of Jensen's Inequality, due to Helson and Lowdenslager [16, Theorem 2]. An independent proof based on the ideas of this section is given in [3]. For all
Let us continue with the proof of Theorem 5.2. We may suppose that is a mean zero trigonometric polynomial, and that the spectrum of is contained in , that is to say
Proof of Theorem 5.1. Transferring inequality (34) by using Theorem 1.8, we obtain that for any set of indices less than , and for any numbers ( ), there is a positive constant , depending only upon the representation , such that
(41) |
Now suppose that is a countable collection of indices less than . Then by Bessaga and Peczynski [5], the series is unconditionally convergent. In particular, for any , for only finitely many do we have that . Since this is true for all such countable sets, we deduce that the set of for which is countable.
Hence we have that is unconditionally convergent to some measure, say . Clearly is weakly measurable. To prove that , it is enough by Proposition 1.4 to show that for every , we have for almost all .
We first note that since for every the series converges to in , it follows that, for every , the series converges to in the weak-* topology of . Now on the one hand, for and , we have , because of the (unconditional) convergence of the series to . On the other hand, by considering the function , we have that , weak *. Thus for almost all , and the proof is complete.