Nicolas A. Tserpes

Idempotent measures on locally compact semigroups

The structure of the support F of an idempotent probability measure Μ on a locally compact semigroup S is considered. It is shown that if S satisfies the condition (L): AB −1 is compact whenever A and B are compact subsets of S, then F is a completely simple semigroup and has the canonical representation X ×G×Y of which G and Y are compact. Moreover, Μ is a product measure Μ X ×Μ G ×Μ Y where Μ X and Μ Y are probability measures and Μ G is the Haar measure on the group G. We conjecture that a similar result remains true even without the condition (L). We give also a relation between our conjecture and a conjecture of Argabright on the support of an r *-invariant measure.

Invariant measures on semigroups with closed translations

where tx denotes the (continuous) right translation s ~ s x . Let Cb(S) and Co(S) be the space of all real valued continuous functions on S which are bounded and have compact supports respectively. For f e C b (S), fx denotes the function fo tx and for B ~ S, CB (x) denotes the characteristic function of B. We denote by # ~ #(. x 1) the measure #x (B) = # (B x 1), for B ~ N. (Similarly for ~# (B) = # (x1 B).) If # is a regular probability measure, Px is also a regular probability measure and the function of x, # (B x 1), for fixed B e N, is (Borel) measurable. In fact, for open V~ S, V~={xeS; #(Vx-~)>a} is open so that the function #(Vx-1), for fixed open V, is lower semicontinuous. (See [3] where the proofs of these facts are given. Their proof applies without modification to our case.) If # and v are regular probability measures, their convolution is defined as the probability measure on N defined by

On semi-invariant probability measures on semigroups

SummaryLet a regular Borel measure m on a locally compact semigroup S be upper semi-invariant i.e., m(C x)≧m(C) and m(x C)≧m(C) for every compact C and x in S. It is shown: (i) Every subsemigroup of S of positive measure contains an idempotent. (ii) S admits an upper semi-invariant probability measure iff S has a kernel K which is a compact group.