Inductive Volume Measure: A Geometric Alternative to Hausdorff Measure

[2507.06540] Inductive Volume Measure: A Geometric Alternative to Hausdorff Measure Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Functional Analysis arXiv:2507.06540 (math) [Submitted on 9 Jul 2025 ( v1 ), last revised 9 Apr 2026 (this version, v4)] Title: Inductive Volume Measure: A Geometric Alternative to Hausdorff Measure Authors: Luis A. Cedeño-Pérez View a PDF of the paper titled Inductive Volume Measure: A Geometric Alternative to Hausdorff Measure, by Luis A. Cede\~no-P\'erez View PDF HTML (experimental) Abstract: We employ methods of geometry and generalized convergence to construct a geometric measure that serves as an alternative to the integer-dimension Hausdorff measure. This construction prioritizes integration, yields the Area Formula as a byproduct of the construction and the Coarea Formula follows indirectly from the smooth case. Furthermore, this is the smallest measure that satisfies the Area Formula. Though this construction is not as general as Hausdorff measure, it provides a much simpler introduction to the topic and is enough for certain applications to Geometric Analysis. Subjects: Functional Analysis (math.FA) ; Differential Geometry (math.DG) Cite as: arXiv:2507.06540 [math.FA] (or arXiv:2507.06540v4 [math.FA] for this version) https://doi.org/10.48550/arXiv.2507.06540 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Luis Antonio Cedeño Pérez [ view email ] [v1] Wed, 9 Jul 2025 04:50:43 UTC (183 KB) [v2] Mon, 14 Jul 2025 21:15:20 UTC (183 KB) [v3] Fri, 20 Mar 2026 18:07:49 UTC (88 KB) [v4] Thu, 9 Apr 2026 19:31:09 UTC (86 KB) Full-text links: Access Paper: View a PDF of the paper titled Inductive Volume Measure: A Geometric Alternative to Hausdorff Measure, by Luis A. Cede\~no-P\'erez View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2025-07 Change to browse by: math math.DG References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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