zbMATH — the first resource for mathematics

On \(\sigma\)-porous sets in abstract spaces. (English) Zbl 1098.28003
This is a survey paper whose aim is to give basic information about properties and applications of \(\sigma\)-porous sets in Banach spaces and some other infinite-dimensional spaces. There are different definitions of porosity in the literature. To distinguish between them the author suggests to use the term “\(\sigma\)-upper porous sets” for “\(\sigma\)-porous sets” as an opposite to “\(\sigma\)-lower porous sets”. Several subsystems of the system of \(\sigma\)-porous sets which have applications in abstract spaces are considered.
These are some subsystems of the system of \(\sigma\)-upper porous sets such as \(\sigma\)-directionally porous sets, \(\sigma\)-strongly porous sets, \(\sigma\)-closure porous sets, etc., and some subsystems of the systems of \(\sigma\)-lower porous sets, namely, cone (angle) small sets, ball small sets, sets covered by surfaces of finite codimension, \(\sigma\)-cone-supported sets, and HP-small sets. Some discussion on smallness in the sense of measure and on descriptive properties for these subsystems follows. If \(X\) is a separable infinite-dimensional Banach space, then three notions of smallness are considered: Gauss null sets, Haar null sets, and \(\Gamma\)-null sets. The applications of \(\sigma\)-porous sets concern differentiability of functions, approximation properties of sets, and variational principles, mainly.
The paper can be considered as a continuation of the author’s survey [Real Anal. Exch. 13, No. 2, 314–350 (1988; Zbl 0666.26003)] and it presents some information on solutions of questions stated in the previous survey as well as an information on the progress made since then in the finite-dimensional case. Only a small number of theorems with almost no proofs are stated in the paper but the list of references is quite long.

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
28A33 Spaces of measures, convergence of measures
28A78 Hausdorff and packing measures
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
46B25 Classical Banach spaces in the general theory
46G12 Measures and integration on abstract linear spaces
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
Zbl 0666.26003
Full Text: DOI EuDML