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    "title": "Hausdorff: measuring sets with fractional and finer dimensions",
    "summary": "Hausdorff extended small-scale covering measures to fractional powers and more general gauge functions, and constructed perfect sets with positive finite measure at prescribed scales under explicit assumptions.\n\nLength, area and volume work well for familiar shapes, but many sets have zero ordinary length or occupy a size between familiar integer dimensions. Hausdorff attached a cost to small covering sets and minimized the total cost as the permitted cover size shrank. A power cost provides a fractional scale; more general functions distinguish sets even when one power exponent does not. Building on Caratheodory’s outer-measure framework, he constructed Cantor-type sets with prescribed positive finite gauge measure under specific concavity conditions. The ternary Cantor set appears at log2/log3, and carefully constrained products support further examples. The original notion of exact gauge dimension is finer than the later common single critical exponent. It is neither topological dimension nor an automatic box-counting estimate. General metric-space formulations, a general gauge-existence theorem, later product theorems and all fractal applications are subsequent contributions. The article is dated March1918 at its end, while the library volume and author-edition bibliography identify publication in1919.",
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        "title": "Felix Hausdorff, Dimension und äußeres Maß — original1919 article, datedMarch1918",
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      "scope": {
        "description": "The construction uses specified covers, limiting diameters and gauge functions to measure sets and define dimension. A gauge choice is part of the object. Dimension need not equal a finite nonzero measure at the critical exponent, and this is not a universal physical scaling law. Later generalized measure/dimension results remain separate.",
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        {
          "citation": "Felix Hausdorff, Dimension und äußeres Maß — original1919 article, datedMarch1918",
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      "statement": "# Hausdorff: measuring sets with fractional and finer dimensions\n\nHausdorff extended small-scale covering measures to fractional powers and more general gauge functions, and constructed perfect sets with positive finite measure at prescribed scales under explicit assumptions.\n\nLength, area and volume work well for familiar shapes, but many sets have zero ordinary length or occupy a size between familiar integer dimensions. Hausdorff attached a cost to small covering sets and minimized the total cost as the permitted cover size shrank. A power cost provides a fractional scale; more general functions distinguish sets even when one power exponent does not. Building on Caratheodory’s outer-measure framework, he constructed Cantor-type sets with prescribed positive finite gauge measure under specific concavity conditions. The ternary Cantor set appears at log2/log3, and carefully constrained products support further examples. The original notion of exact gauge dimension is finer than the later common single critical exponent. It is neither topological dimension nor an automatic box-counting estimate. General metric-space formulations, a general gauge-existence theorem, later product theorems and all fractal applications are subsequent contributions. The article is dated March1918 at its end, while the library volume and author-edition bibliography identify publication in1919.\n\n## Scientific scope\n\nThe construction uses specified covers, limiting diameters and gauge functions to measure sets and define dimension. A gauge choice is part of the object. Dimension need not equal a finite nonzero measure at the critical exponent, and this is not a universal physical scaling law. Later generalized measure/dimension results remain separate.\n\n## People and systems\n\nFelix Hausdorff\n\n\n\n## Original sources\n\n- [Felix Hausdorff, Dimension und äußeres Maß — original1919 article, datedMarch1918](https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0079/LOG_0013.pdf)\n- [S.D.Chatterji, collected-edition Commentary on H1919a — original/later-scope distinction](https://hausdorff-edition.de/media/pdf/Kommentar_H1919a.pdf)\n",
      "summary": "Hausdorff extended small-scale covering measures to fractional powers and more general gauge functions, and constructed perfect sets with positive finite measure at prescribed scales under explicit assumptions.\n\nLength, area and volume work well for familiar shapes, but many sets have zero ordinary length or occupy a size between familiar integer dimensions. Hausdorff attached a cost to small covering sets and minimized the total cost as the permitted cover size shrank. A power cost provides a fractional scale; more general functions distinguish sets even when one power exponent does not. Building on Caratheodory’s outer-measure framework, he constructed Cantor-type sets with prescribed positive finite gauge measure under specific concavity conditions. The ternary Cantor set appears at log2/log3, and carefully constrained products support further examples. The original notion of exact gauge dimension is finer than the later common single critical exponent. It is neither topological dimension nor an automatic box-counting estimate. General metric-space formulations, a general gauge-existence theorem, later product theorems and all fractal applications are subsequent contributions. The article is dated March1918 at its end, while the library volume and author-edition bibliography identify publication in1919.",
      "title": "Hausdorff: measuring sets with fractional and finer dimensions"
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    "title": "Hausdorff: measuring sets with fractional and finer dimensions",
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    "what_changed": "Hausdorff extended small-scale covering measures to fractional powers and more general gauge functions, and constructed perfect sets with positive finite measure at prescribed scales under explicit assumptions.",
    "scope": "Hausdorff extended small-scale covering measures to fractional powers and more general gauge functions, and constructed perfect sets with positive finite measure at prescribed scales under explicit assumptions. The construction uses specified covers, limiting diameters and gauge functions to measure sets and define dimension. A gauge choice is part of the object. Dimension need not equal a finite nonzero measure at the critical exponent, and this is not a universal physical scaling law. Later generalized measure/dimension results remain separate.",
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