Ultrafilters on measurable semigroups

Abstract

Let $(S,\cdot)$ be a semigroup and $\mathfrak{m}$ be a $\sigma$-algebra on $S$. We say $(S,\cdot,\mathfrak{m})$ is a measurable semigroup if $\pi:S\times S\longrightarrow S$ by $\pi(x,y)=x\cdot y$ is a measurable function. In this paper , we consider to $\mathfrak{m}^\beta$ as the collection of all ultrafilters on $\mathfrak{m}$. We show that $\mathfrak{m}^\beta$ is a compact right topological semigroup respect to generated topology by $\sigma$-algebra $\mathfrak{m}$ on $\mathfrak{m}^\beta$. Also we study some elementary properties of the algebraic structure of $\mathfrak{m}^\beta$.