Measurable centres in convolution semigroups

TL;DR

This paper proves that in convolution semigroups over locally compact groups, measurable translation elements are necessarily continuous, showing the measurable centre and topological centre are the same.

Contribution

It establishes a fundamental equivalence between measurability and continuity for translation elements in convolution semigroups over locally compact groups.

Findings

Measurable translation elements are continuous in convolution semigroups.

The measurable centre equals the topological centre in these semigroups.

Continuity of translation is guaranteed by measurability in this context.

Abstract

In a convolution semigroup over a locally compact group, measurability of the translation by a fixed element implies continuity. In other words, the measurable centre coincides with the topological centre.