Proof of the Ghahramani-Lau conjecture

•The measure algebra of every locally compact group is strongly Arens irregular.•Separation of measures by translation studied on arbitrary locally compact groups.•New subalgebras of measures on locally compact groups are introduced and studied.•Strong Arens irregularity is also established for the case of Polish groups.

The Ghahramani-Lau conjecture is established; in other words, the measure algebra of every locally compact group is strongly Arens irregular. To this end, we introduce and study certain new classes of measures (called approximately invariant, respectively, strongly singular) which are of interest in their own right. Moreover, we show that the same result holds for the measure algebra of any (not necessarily locally compact) Polish group.