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