Geodesic
Note on the Borel Method of Measure Extension
Canadian mathematical bulletin, Tome 5 (1962) no. 3, pp. 285-296

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DOI

This note concerns a countably additive measure on a Boolean ring of subsets of an abstract set, this measure being real-valued, admitting ∞ as a possible value. We are interested only in unique extensions, so we suppose the measure to be σ- finite. The following well known result will be referred to as the "extension theorem": "Every σ-finite measure on a ring extends uniquely to a σ-finite measure on the generated σ-ring. "Besides the familiar proof using outer measure, there is a Borel-type proof using transfinite induction [4]. We attempt here to reduce the Borel-type proof to its ultimate simplicity, reducing the problem to the bounded case. 
Fox, G. Note on the Borel Method of Measure Extension. Canadian mathematical bulletin, Tome 5 (1962) no. 3, pp. 285-296. doi: 10.4153/CMB-1962-029-6
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     title = {Note on the {Borel} {Method} of {Measure} {Extension}},
     journal = {Canadian mathematical bulletin},
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     year = {1962},
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     number = {3},
     doi = {10.4153/CMB-1962-029-6},
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