{
  "abstract": "We study strong measurability and adapted approximations for maps taking values in possibly nonseparable metric spaces. After recalling the metric-space Pettis criterion, we give conditions under which measurability implies strong measurability. Under the continuum hypothesis (CH), every measurable map from a countably generated measurable space into a metric space is strongly measurable for every measure on the domain. For finite or $σ$-finite measures, the same conclusion holds when the target density is strictly smaller than every real-valued measurable cardinal. Under CH this includes targets of density at most $\\mathfrak c$; if no real-valued measurable cardinal exists, it holds for all metric targets. For function-valued maps, we show that pointwise measurability and continuity of the sections need not ensure measurability in the function-space topology. We provide a verification criterion using countably many evaluations and essential separability of the range. Further, we obtain elementary $L^p$-approximations of progressive or jointly measurable adapted processes under suitable filtration assumptions, Lipschitz functionals with path and Wasserstein variables, and smooth approximations based on finitely many observations of a process with left- or right-continuous paths. Finally, under CH and suitable assumptions, we apply these results to controlled stochastic differential equations in separable Hilbert spaces, allowing for jumps, random coefficients, path dependence, and state- and control-law dependence; we establish convergence of states and costs uniformly over admissible controls, convergence of optimal values, and transfer of near-optimal controls.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27478",
  "author_total": 2,
  "authors": [
    {
      "name": "Keivan Mirzaei"
    },
    {
      "name": "Jinniao Qiu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/91j3mkejwf3j2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27478",
  "fulltext_url": null,
  "id": "91j3mkejwf3j2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27478",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.27478",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/91j3mkejwf3j2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27478",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27478"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Strong Measurability and Adapted Approximations with Applications to Non-Markovian Control",
  "title_text": "Strong Measurability and Adapted Approximations with Applications to Non-Markovian Control",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/91j3mkejwf3j2",
  "venue": "arXiv",
  "versions": []
}