{
  "abstract": "Let $g$ be a nondecreasing left-continuous function and let $μ_g$ be its Lebesgue--Stieltjes measure. We establish a Banach-valued fundamental theorem of calculus and an Aubin--Lions compactness theorem for evolution measured by $μ_g$, allowing absolutely continuous, singular continuous and atomic components. If the range space has the Radon--Nikodým property, a curve is $g$-absolutely continuous if and only if it is an indefinite Bochner integral; its strong $g$-derivative is the Bochner density and the variation measure has density equal to its norm. A terminal atom may make the derivative invisible from the Bochner state class, so the natural evolution space is a graph of state--derivative pairs. For $B_0\\Subset B\\hookrightarrow B_1$, with $B_0$ and $B_1$ reflexive and $1<p_0,p_1<\\infty$, the state projection is a compact linear operator into $L_g^{p_0}([a,b);B)$. The proof combines bounded evaluation at atoms, local $μ_g$-averages at nonatomic points and Ehrling's inequality. The result recovers the classical theorem and yields compactness principles for bounded time scales, weighted sequences and mixed Stieltjes measures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10064",
  "author_total": 1,
  "authors": [
    {
      "name": "Francisco J. Fernández"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d3pb7z7130vj9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10064",
  "fulltext_url": null,
  "id": "d3pb7z7130vj9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10064",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7212138755",
  "pdf_url": "https://arxiv.org/pdf/2609.10064",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d3pb7z7130vj9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.10064",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10064"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Aubin--Lions Compactness for Stieltjes--Bochner Evolution Graphs",
  "title_text": "Aubin--Lions Compactness for Stieltjes--Bochner Evolution Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/d3pb7z7130vj9",
  "venue": "arXiv",
  "versions": []
}