A note on conservative measures on semigroups

Abstract

Consider (S,B,μ) the measure space where S is a topological metric semigroup and μ a countably additive bounded Borel measure. Call μ conservative if all right translations tx:s→sx, x∈S (which are assumed closed mappings) are conservative with respect (S,B,μ) in the ergodic theory sense. It is shown that the semigroup generated by the support of μ is a left group. An extension of this result is obtained for σ-finite μ.

Keywords

• topological metric semigroups
• left groups
• conservative translations of the semigroup
• Borel measure
• r*-invariant measures.