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The interior operation in a \(\psi\)-density topology. (English) Zbl 0949.26002

Let \(\psi :(0,\infty)\to (0,\infty)\) be a continuous non-decreasing function such that \(\lim_{x\rightarrow 0^+}\psi (x)=0\). A point \(x \in {\mathcal R}\) is a \(\psi \)-density point of a measurable set \(A \subset {\mathcal R}\) if \(\lim_{h \rightarrow 0^+}\frac{m(A'\cap [x-h,x+h])}{2h\psi (2h)} = 0\). The family of all measurable sets such that each point \(x \in A\) is a \(\psi \)-density point of \(A\) is a topology called \(\psi \)-topology. In this paper it is proved that for an arbitrary set \(A\) its interior in a \(\psi \)-topology is equal to \(A\cap \varphi ^{\beta }(B)\), where \(B\) is a measurable kernel of \(A\) and \(\beta \) is some countable ordinal. Moreover, each countable ordinal \(\beta \geq 1\) realizes the interior of some measurable set \(A\).

MSC:

26A03 Foundations: limits and generalizations, elementary topology of the line
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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References:

[1] Oxtoby, J. C., Measure and category (1971), New York: Springer Verlag, New York · Zbl 0217.09201
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[3] Taylor, S. J., On strengthening the Lebesgue Density Theorem, Fund. Math., 46, 305-315 (1959) · Zbl 0086.04601
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