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**Topics on analytic sets.**
*(English)*
Zbl 0764.28002

The paper concerns some relations between analysis and descriptive set theory.

An alternative proof of a recent result of Kechris and Lyons is indicated. It claims CA measurability and Borel nonmeasurability of probability measures annihilating all Rajchman sets in the circle \(T\) of length \(2\pi\) which are related to a fixed infinite \(M\subset N\) and an open non-empty \(V\subset T\).

The second topic are PCA subsets of \(C[0,1]\). A Becker’s theorem is improved to the fact that PCA subsets of \(C[0,1]\) can be represented as the sets of pointwise limit points of subsequences of \((f_ n)\) with all (pointwise) convergent subsequences uniformly bounded.

The rest is devoted to norm attaining functionals: Theorem \({\mathcal A}\) says: Let \(X\) be a separable nonreflexive Banach space. There is an equivalent norm on \(X\) producing a non-Borel set of norm attaining functionals. A variant for the dual norm is also proved.

An alternative proof of a recent result of Kechris and Lyons is indicated. It claims CA measurability and Borel nonmeasurability of probability measures annihilating all Rajchman sets in the circle \(T\) of length \(2\pi\) which are related to a fixed infinite \(M\subset N\) and an open non-empty \(V\subset T\).

The second topic are PCA subsets of \(C[0,1]\). A Becker’s theorem is improved to the fact that PCA subsets of \(C[0,1]\) can be represented as the sets of pointwise limit points of subsequences of \((f_ n)\) with all (pointwise) convergent subsequences uniformly bounded.

The rest is devoted to norm attaining functionals: Theorem \({\mathcal A}\) says: Let \(X\) be a separable nonreflexive Banach space. There is an equivalent norm on \(X\) producing a non-Borel set of norm attaining functionals. A variant for the dual norm is also proved.

Reviewer: P.Holicky (Praha)

### MSC:

28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |

54H05 | Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) |

46B10 | Duality and reflexivity in normed linear and Banach spaces |

28A33 | Spaces of measures, convergence of measures |

42A99 | Harmonic analysis in one variable |

46E15 | Banach spaces of continuous, differentiable or analytic functions |