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A relationship between intersection conditions and porosity conditions for local systems. (English) Zbl 0793.26007
In this paper the author proves that if a local system \(\mathbb{S}\) fulfils the conditions: 1. \(\mathbb{S}(x)= \{x+ S;\;S\in\mathbb{S}(0)\}\) for every \(x\), 2. \(S^ +\cup- S^ +\in\mathbb{S}(0)\) and \(S^ -\cup- S^ -\in \mathbb{S}(0)\) whenever \(S\in\mathbb{S}(0)\), 3. If \(\mathbb{S}\) satisfies the parametric intersection condition, then, for any \(x\in\mathbb{R}\) and \(S\in \mathbb{S}(x)\), \[ p^ +(S,x)\leq {(1/2)+\lambda\over 1+\lambda}\quad\text{and}\quad p^ -(S,x)\leq {(1/2)+\lambda\over 1+\lambda} \] (i.e., the porosity of sets from \(\mathbb{S}(x)\) at \(x\) cannot be too large).
Reviewer: R.Pawlak (Łódź)
MSC:
26A21 Classification of real functions; Baire classification of sets and functions
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References:
[1] THOMSON B. S.: Real Functions. Springer-Verlag, 1985. · Zbl 0581.26001
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