On \(\sigma\)-porous sets in abstract spaces.

*(English)*Zbl 1098.28003This 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.

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.

Reviewer: Miroslav Repický (Košice)

##### MSC:

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) |