# zbMATH — the first resource for mathematics

Semistable convolution semigroups on measurable and topological groups. (English) Zbl 0544.60021
Semistable probability measures on the real line have been first considered by P. Lévy. The extension to operator-semistable laws on Euclidean spaces is due to R. Jajte. Far reaching generalizations to topological and measurable groups have been studied by A. Tortrat [see e.g., Probability measures on groups, Proc. 6th Conf., Oberwolfach 1981, Lect. Notes Math. 928, 452-466 (1982; Zbl 0514.60014)]. Following this approach we adopt in this paper the following definition: A convolution semigroup (c.s.g.) $$(\mu_ t)_{t>0}$$ of probability measures on a measurable group G is said to be (strictly) semistable with respect to a measurable homomorphism $$\delta$$ of G and with coefficient $$c\in(0,1)\cup(1,\infty)$$ if $$\delta(\mu_ t)=\mu_{ct}$$ for all $$t>0.$$
Several examples from the literature that fit into this framework are presented in section 1. In section 2 a rather general result on purity of semistable c.s.g. is proved (theorem 1): These semigroups respect certain band decompositions of the Banach lattice of bounded measures on G. On the one hand side this leads to the well known zero-one laws for semistable measures and on the other hand side to the result that semistable c.s.g. on locally compact groups are of pure Lebesgue type. In section 3 we show that under rather weak conditions a semistable c.s.g. is holomorphic (theorem 5). This extends a well known property of stable c.s.g. also due to the author [Holomorphic convolution semigroups on topological groups. Probability measures on groups VII. Proc. Conf., Oberwolfach 1983. Lect. Notes Math. 1064, 421-449 (1984)]. In the semistable case the proof is more complicated; one first has to establish the quasianalyticity of the c.s.g. (theorem 4). In section 4 the results of section 3 are specialized to Euclidean spaces. In particular the support structure of a semistable c.s.g. is investigated (theorem 7).
Remark: It has been incorrectly stated in our paper that a topological Hausdorff group G is also a measurable group (cf. p. 150). (This holds if G has a countable basis of its topology.) Nevertheless all of our results hold true: The product measure $$\mu \otimes \nu$$ can be uniquely extended to a $$\tau$$-regular signed measure $$\mu {\hat \otimes}\nu$$ on $${\mathcal B}(G\times G)$$, and Fubini’s theorem remains valid for $$\mu {\hat \otimes}\nu$$ [cf. I. Csiszár, Probab. Struct. algébriques, Actes Colloque internat. 186, Clermont-Ferrand 1969, 75-97 (1970; Zbl 0229.60004)].

##### MSC:
 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 43A05 Measures on groups and semigroups, etc.
Full Text: