Probabilities and Baire's theory in harmonic analysis (Q2772768)

This is an excellent expository paper, with an emphasis on the history of the subject. Without entering into details, we try to convey some of its flavour to the interested reader.NEWLINENEWLINENEWLINEIn the terminology of Baire, the countable unions of nowhere dense sets are called sets of first category. In Bourbaki's terminology, they are called meager sets. They are the analogues of null sets in measure theory. The complements of sets of the first category are called sets of the second category. They are the analogues of sets of full measure, and they can be defined as sets that contain a dense \(G_\delta\) set. When a property holds on a dense \(G_\delta\) set, it is called generic or quasi sure. This is the analogue of almost sure in probability theory. One may also say that the property holds quasi everywhere, the analogue of almost everywhere in the theory of real functions.NEWLINENEWLINENEWLINEHowever, it is important to emphasize that almost sure and quasi sure properties can be very different, that null sets can be sets of the second category, and that sets of full meaure can be meager sets.NEWLINENEWLINENEWLINEThe paper consists of three main parts: (i) history, terminology, and examples (Sections 1 and 2), (ii) functions and series (Sections 3-5), (iii) thin sets (Sections 6-10).NEWLINENEWLINENEWLINEThe titles of the Sections are the following: 1. Lebesgue, Baire, and trigonometric series, 2. Almost sure and quasi sure; examples, 3. Nowhere-differentiable functions, 4. Random Taylor series: continuation, convergence, and divergence, 5. Generic trigonometric and power series, 6. Thin sets and function spaces, 7. Sidon and Zygmund sets, 8. Kronecker, Helson, \(M\), and Salem sets, 9. Random thin sets, 10. Generic thin sets.NEWLINENEWLINENEWLINEThe References contain 57 items.NEWLINENEWLINENEWLINEThis is a very clearly written, enjoyable paper. We recommend it to everybody who would like to get up-to-date information at first hand on the latest developments in harmonic analysis.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00019].