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A Borel measure ,u on a compact abelian group G is LP-improving if, given p > 1, there is a q = q(p,,u) > p and a K = K(p,q,,u) > 0 such that IlJU * fIIq 1, there is a q = q(p, ,) > p and a K = K(p, q, ,) > 0 such that ,u satisfies (1) itA * fllq < KllflIp for each f E LP(G). In [2], Bonami showed that particular Riesz product measures defined on the circle, on the countable product of circles, and on the Cantor group are LP-improving. Besides providing quantitative results regarding convolution against these measures, she gave a sufficient condition for certain Riesz products defined on an arbitrary compact abelian group G to be LP-improving. What she did not do, however, was to characterize the LP-improving Riesz product measures although a general definition of Riesz product due to Hewitt and Zuckerman was in the literature at that time [6]. Our point of departure in this paper is Bonami's work. Here, occasioned by the existence of the more encompassing question raised by Stein in [8], we characterize the LP-improving Riesz product measures. Before we do this, though, we recall the definition of the set R(G) of Riesz product measures defined on G in terms of infinite dissociate subsets of F. In recounting the definition, we follow [3] or [5, Ch. 7], but we use additive notation for the group operation on F. A subset 0 of F is said to be dissociate if each in F can be written uniquely, except possibly for the order of the summands, in the form

Most Riesz product measures are 𝐿^{𝑝}-improving