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Let $m$ be an additive function on a ring $H$ of sets, with values in a commutative Hausdorff topological group, and let $K$ be an ideal of $H$. Conditions are given under which $m$ can be represented as the sum of two additive functions, one essentially supported on $K$, the other vanishing on $K$. The result is used to obtain two Lebesgue-type decomposition theorems. Other applications and the corresponding theory for outer measures are also indicated.