An s-bounded additive map on
a topological Boolean algebra to a topological group can be decomposed into a
continuous and a singular part. This can be done in a canonical way as a limit
thoerem in spaces of operators. As a consequence, if 𝒜 is a Boolean algebra
of continuous projections on a (complete) topological group X and 𝒢 is
a “Fréchet-Nikodým” topology on 𝒜, then every x in X, viewed as an
additive map A → Ax on 𝒜, can be decomposed uniquely as the sum of a
𝒢-continuous and a 𝒢-singular part. If 𝒜 is equicontinuous, the operators which
decompose x are continuous. The result applies to the space of all s-bounded
additive functions on an algebra of sets to a complete separated topological
group.
