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- W2898841705 abstract "Abstract For the σ -ideal <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>N</m:mi> </m:math> $mathcal{N} $ of nullsets and σ -ideal <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>M</m:mi> </m:math> $mathcal{M} $ of microscopic sets, it was recently obtained that there exists a Borel set <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>E</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> $Esubsetmathbb{R}^{2} $ with the following property: <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>E</m:mi> <m:mi>x</m:mi> </m:msub> <m:mo>∈</m:mo> <m:mi mathvariant=script>M</m:mi> </m:mrow> </m:math> $E_{x}inmathcal{M} $ for any <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> $xinmathbb{R} $ and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mi>y</m:mi> <m:mo>;</m:mo> <m:msup> <m:mi>E</m:mi> <m:mi>y</m:mi> </m:msup> <m:mo>∉</m:mo> <m:mi mathvariant=script>N</m:mi> <m:mo stretchy=false>}</m:mo> </m:mrow> <m:mo>∉</m:mo> <m:mi mathvariant=script>M</m:mi> </m:mrow> </m:math> ${y;E^{y}notinmathcal{N}}notinmathcal{M} $ , for vertical sections <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>E</m:mi> <m:mi>x</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mi>y</m:mi> <m:mo>;</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:mi>E</m:mi> <m:mo stretchy=false>}</m:mo> </m:mrow> </m:mrow> </m:math> $E_{x}={y;(x,y)in E} $ and horizontal sections <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>E</m:mi> <m:mi>y</m:mi> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mi>x</m:mi> <m:mo>;</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:mi>E</m:mi> <m:mo stretchy=false>}</m:mo> </m:mrow> </m:mrow> </m:math> $E^{y}={x;(x,y)in E} $ for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>E</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> $Esubsetmathbb{R}^{2} $ . Thus <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi mathvariant=script>N</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=script>M</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:math> $(mathcal{N},mathcal{M}) $ does not satisfy Fubini Property. In this paper we obtain such Borel set E , that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mi>y</m:mi> <m:mo>;</m:mo> <m:msup> <m:mi>E</m:mi> <m:mi>y</m:mi> </m:msup> <m:mo>∉</m:mo> <m:mi mathvariant=script>N</m:mi> <m:mo stretchy=false>}</m:mo> </m:mrow> <m:mo>∉</m:mo> <m:mi mathvariant=script>M</m:mi> </m:mrow> </m:math> ${y;E^{y}notinmathcal{N}}notinmathcal{M} $ and all non-horizontal (in a natural sense) sections of E are in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>M</m:mi> </m:math> $mathcal{M} $ . Other Fubini type properties, with conditions written for all directions are also discussed." @default.
- W2898841705 created "2018-11-09" @default.
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- W2898841705 date "2018-10-01" @default.
- W2898841705 modified "2023-09-27" @default.
- W2898841705 title "On microscopic sets and Fubini Property in all directions" @default.
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- W2898841705 doi "https://doi.org/10.1515/ms-2017-0165" @default.
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