CONTENTS Introduction..............................................................................................................5 1. Preliminaries........................................................................................................6 1.0. Measurable, probabilistic, and statistical spaces..............................................6 1.1. Transition functions..........................................................................................6 1.2. Linear space of P-bounded measurable real functions....................................8 1.3. Almost transition functions................................................................................9 2. Concrete categories of measurable, probabilistic, and statistical spaces..........14 3. Categories connected with transition functions..................................................16 3.1. Category $Mes_{trans}$.................................................................................16 3.2. Category $St_{trans}$....................................................................................19 3.3. Category $St_{dist}$......................................................................................20 3.4. Categories of indexed statistical spaces.........................................................23 4. Categories connected with almost transition functions......................................23 Final remarks.........................................................................................................27 References............................................................................................................28
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We prove that if $ϱ_H$ and δ are the Hausdorff metric and the radial metric on the space 𝓢ⁿ of star bodies in ℝ, with 0 in the kernel and with radial function positive and continuous, then a family 𝓐 ⊂ 𝓢ⁿ that is meager with respect to $ϱ_H$ need not be meager with respect to δ. Further, we show that both the family of fractal star bodies and its complement are dense in 𝓢ⁿ with respect to δ.
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In 1989 R. Arnold proved that for every pair (A,B) of compact convex subsets of ℝ there is an Euclidean isometry optimal with respect to L₂ metric and if f₀ is such an isometry, then the Steiner points of f₀(A) and B coincide. In the present paper we solve related problems for metrics topologically equivalent to the Hausdorff metric, in particular for $L_p$ metrics for all p ≥ 2 and the symmetric difference metric.
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In 1989 R. Arnold proved that for every pair (A,B) of compact convex subsets of ℝ there is an Euclidean isometry optimal with respect to L₂ metric and if f₀ is such an isometry, then the Steiner points of f₀(A) and B coincide. In the present paper we solve related problems for metrics topologically equivalent to the Hausdorff metric, in particular for $L_p$ metrics for all p ≥ 2 and the symmetric difference metric.
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