Geodesic
Inductive Extension of a Vector Measure Under a Convergence Condition
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 1246-1255

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Let μ be a vector measure (countably additive set function with values in a Banach space) on a field. If μ is of bounded variation, it extends to a vector measure on the generated σ-field (2; 5; 8). Arsene and Strătilă (1) have obtained a result, which when specialized somewhat in form and context, reads as follows: “A vector measure on a field, majorized in norm by a positive, finite, subadditive increasing set function defined on the generated σ-field, extends to a vector measure on the generated σ-field”. 
Fox, Geoffrey. Inductive Extension of a Vector Measure Under a Convergence Condition. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 1246-1255. doi: 10.4153/CJM-1968-120-x
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