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<p>In this note we prove that the values of the lower density operator introduced by W. Wilczyński in 2015 are Borel sets of the class <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2455_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Pi _{14}^0\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Π</mi> <mrow> <mn>14</mn> </mrow> <mn>0</mn> </msubsup> </math></EquationSource> </InlineEquation>. We show also that there exists a lower density operator on the plane with non-Borel values and that there exist lower density operators on the plane with values from the arbitrary Borel level.</p>

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Lower Density Operators on the Plane with Borel and Non-Borel Values

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月18日

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  • Gertruda Ivanova,
  • Elżbieta Wagner-Bojakowska

摘要

In this note we prove that the values of the lower density operator introduced by W. Wilczyński in 2015 are Borel sets of the class \(\Pi _{14}^0\) Π 14 0 . We show also that there exists a lower density operator on the plane with non-Borel values and that there exist lower density operators on the plane with values from the arbitrary Borel level.