{
  "id": "1712.05212",
  "version": "v1",
  "published": "2017-12-14T13:19:21.000Z",
  "updated": "2017-12-14T13:19:21.000Z",
  "title": "Nonmeasurable sets and unions with respect to tree ideals",
  "authors": [
    "Marcin Michalski",
    "Robert Rałowski",
    "Szymon Żeberski"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "In this paper we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals $s_0$, $m_0$, $l_0$, and $cl_0$. We show that there exists a subset $A$ of the Baire space $\\omega^\\omega$ which is $s$-, $l$-, and $m$-nonmeasurable, that forms dominating m.e.d. family. We introduce and investigate a notion of $\\mathbb{T}$-Bernstein sets - sets that intersect but does not containt any body of a tree from a given family of trees $\\mathbb{T}$. We also acquire some results on $\\mathcal{I}$-Luzin sets, namely we prove that there are no $m_0$-, $l_0$-, and $cl_0$-Luzin sets and that if $\\mathfrak{c}$ is a regular cardinal, then the algebraic sum (considered on the real line $\\mathbb{R}$) of a generalized Luzin set and a generalized Sierpi\\'nski set belongs to $s_0, m_0$, $l_0$ and $cl_0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-14T13:19:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E17",
      "03E50",
      "03E75"
    ],
    "keywords": [
      "nonmeasurable sets",
      "generalized sierpinski set belongs",
      "baire space",
      "regular cardinal",
      "real line"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}