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

Abstract 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 $$\Pi _{14}^0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mrow> <mml:mn>14</mml:mn> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> . 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.