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Non-linear images of \(\mu\)-shadings, shadings in \(\mathbb{R}^2\), and quotient sets of \(\mu\)-shadings. (English) Zbl 1298.28008

If \(f:\mathbb{R}\to [0,1]\) is continuous then \(A\subset\mathbb{R}\) is called an \(f\)-shading, if for each Banach measure \(\mu\) and \(x\in\mathbb{R}\), \(\lim_{|I|\to 0}\frac{\mu(A\cap I)}{|I|}=f(x)\), when the limit is taken over all bounded intervals which contain \(x\). A set \(A\) is called shading with the shade \(sh(A)=a\in [0,1]\) if \(A\) is an \(f\)-shading for a constant function \(f=a\). See R. D. Mabry [Real Anal. Exch. 16, No. 2, 425–459 (1991; Zbl 0735.28001)]. In the paper under review the authors consider some natural modifications of these notions (e.g., shadings in \(\mathbb{R}^2\) or shadings with respect to t-Banach measures) and algebraic and topological properties of such sets, like continuous images, Cartesian products, or quotient sets. For example, they prove the following theorems. (1) If \(f\) is a continuous and increasing bijection and \(A\) is an almost isometry-invariant shading with the shade \(0\) then \(sh(f(A))=0\). (2) Let \(f,g:\mathbb{R}\to [0,1]\) be continuous functions, \(F\) be an \(f\)-shading on \(\mathbb{R}\) and \(G\) be a \(g\)-shading on \(\mathbb{R}\). Then \(sh(F\times G)(x,y)=f(x)g(x)\). In particular, if \(A,B\) are shadings on \(\mathbb{R}\) then \(A\times B\) is a shading on \(\mathbb{R}^2\) with \(sh(A\times B)=sh(A)\cdots h(B)\). This solves a problem posed by R. Mabry.

MSC:

28A12 Contents, measures, outer measures, capacities

Citations:

Zbl 0735.28001
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