## 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

Zbl 0735.28001
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