Vol. 80, No. 1, 1979

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The group-valued Lebesgue decomposition

Tim Eden Traynor

Vol. 80 (1979), No. 1, 273–277

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.

Mathematical Subject Classification 2000
Primary: 28B10
Received: 8 July 1977
Published: 1 January 1979
Tim Eden Traynor