Measure algebras and functions of bounded variation on idempotent semigroups

Newman, Stephen E.

doi:10.1090/S0002-9947-1972-0308686-4

Measure algebras and functions of bounded variation on idempotent semigroups
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Trans. Amer. Math. Soc. 163 (1972), 189-205 Request permission

Abstract:

Our main result establishes an isomorphism between all functions on an idempotent semigroup $S$ with identity, under the usual addition and multiplication, and all finitely additive measures on a certain Boolean algebra of subsets of $S$, under the usual addition and a convolution type multiplication. Notions of a function of bounded variation on $S$ and its variation norm are defined in such a way that the above isomorphism, restricted to the functions of bounded variation, is an isometry onto the set of all bounded measures. Our notion of a function of bounded variation is equivalent to the classical notion in case $S$ is the unit interval and the “product” of two numbers in $S$ is their maximum.

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