On σ‐porous sets in abstract spaces

Abstract

The main aim of this survey paper is to give basic information about properties and applications ofσ-porous sets in Banach spaces(and some other infinite-dimensional spaces). This paper can be considered a partial continuation of the author's 1987 survey on porosity andσ-porosity and therefore only some results, remarks, and references (important for infinite-dimensional spaces) are repeated. However, this paper can be used without any knowledge of the previous survey. Some new results concerningσ-porosity in finite-dimensional spaces are also briefly mentioned. However, results concerning porosity (but notσ-porosity) are mentioned only exceptionally.

A central limit theorem, and related results, for a two-color randomly reinforced urn

“…. , R X (n + 1) is independent of σ (δ n+1 , A n ) and, thus, Therefore, (20) holds for all n; hence, P(Z ∞ = 0) ≤ P( Z ∞ = 0) = 0. It remains to prove that P(Z ∞ ∈ (0, 1)) = 0.…”

“…Unfortunately, the idea developed in the proof of Theorem 2 cannot be further exploited to produce such a result. In any case, if the closed balls B n appearing in the proof are replaced with the 'holes' of a porous set (for the link between σ -porous sets and measures, see [12] and [20]), it is possible to show that P(Z ∞ ∈ S) = 0 for all σ -porous sets S in [0, 1]. Unfortunately, this is not enough to prove that the distribution of Z ∞ is absolutely continuous; indeed, T -measures are singular with respect to the Lebesgue measure, but they attribute 0-measure to any σ -porous set (see, e.g.…”

Porosity and the L p-conjecture

“…for some n ∈ N and x ∈ K. Hence by (1), Δ(y −1 ) = Δ(a n )Δ(x −1 ) ≤ S. Finally, by (4), (5), (6), and (7), we have…”

Developments in fractal geometry

“…The metric space ( , d ) is defined formally at the beginning of Section 6. A meager set is a set of the first Baire category; the definition of σ -porous can be found in [93,127] and results using porosity in fractal geometry in [46,83]. The definition of a strongly-fibred attractor is given in Sect.…”