Abstract
Measure and integration theory for finitely additive measures, including vector-valued measures,
is shown to be essentially covered by a class of commutative
L-algebras, called measurable algebras. The domain and range of any
measure is a commutative L-algebra. Each measurable algebra embeds
into its structure group, an abelian group with a compatible lattice
order, and each (general) measure extends uniquely to a monotone group
homomorphism between the structure groups. On the other hand, any measurable
algebra X is shown to be the range of an essentially unique measure on
a measurable space, which plays the role of a universal covering. Accordingly,
we exhibit a fundamental group of X, with stably closed subgroups
corresponding to a special class of measures with X as target. All structure
groups of measurable algebras arising in a classical context are archimedean.
Therefore, they admit a natural embedding into a group of extended real-valued
continuous functions on an extremally disconnected compact space, the Stone space of the measurable algebra.
Extending Loomis’ integration theory for
finitely additive measures, it is proved that, modulo null functions, each
integrable function can be represented by a unique continuous function on the Stone space.
The algebra of finitely additive measures on a partially ordered semigroup
