{
  "annotation_id": "8dd53f28-b738-45d3-913d-b1f5903b7b84",
  "date_created": "2026-02-17T05:53:08.707000Z",
  "date_modified": "2026-02-17T05:53:08.707000Z",
  "file_hash": "9ecced48ba590812fafadde0c88bd16f7c4b5c29fe4e56539aa1ca7235f3e464",
  "private": false,
  "record": {
    "abstract": "Let $(X, \\mfA,P)$ and $(Y, \\mfB,Q)$ be two probability spaces, $R$ be their skew product on the product $\\sigma$-algebra $\\mfA\\otimes\\mfB$ and $\\{(\\mfA_y,S_y)\\colon y\\in{Y}\\}$ be a $Q$-disintegration of $R$. Then let $\\mfA\\dd\\mfB$ be the $\\sigma$-algebra generated $\\mfA\\otimes\\mfB$ and by the family $\\mcM:=\\{E\\subset{X\\times{Y}}\\colon \\exists\\;N\\in\\mfB_0\\;\\forall\\;y\\notin{N}\\;\\wh{S_y}(E^y)=0\\}$ and $\\wh{R_{\\dd}}$ be the extension of $R$ such that $\\mcM$ becomes the family of $\\wh{R_*}$-zero sets ($\\wh{S_y}$ is the completion of $S_y$ and $\\mfB_0=\\{B\\in\\mfB: Q(B)=0\\}$). We prove that there exist a lifting $\\pi$ on $\\mcL^{\\infty}(\\wh{R_{\\dd}})$ and liftings $\\sigma_y$ on $\\mcL^{\\infty}(\\wh{S_y})$ , $y\\in Y$, such that \\[ [\\pi(f)]^y= \\sigma_y\\Bigl([\\pi(f)]^y\\Bigr) \\qquad\\mbox{for every} \\quad y\\in Y\\quad\\mbox{and every}\\quad f\\in\\mcL^{\\infty}(\\wh{R_{\\dd}}). \\] In case of a separable $P$ and in case when $R\\ll{P}\\times{Q}$ a characterization of stochastic processes possessing an equivalent measurable version is presented. The theorem is a generalization and correction of \\cite[Theorem 3.8]{mu25}.",
    "arxiv_id": "2601.06538",
    "authors": [
      "Kazimierz Musial"
    ],
    "categories": [
      "math.PR"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "title": "Splitting of Liftings in Product Spaces II",
    "url": "https://arxiv.org/abs/2601.06538",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "b74663f8-0fa7-4ebe-b6f2-772d5361cdc6",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}