Probabilities and Baire’s theory in harmonic analysis, in: Twentieth Century Harmonic Analysis, A Celebration

ABSTRACT. This is an expository paper, with an emphasis on the history of the sub-ject. It consists of three main parts: history, terminology, and examples (Sections 1 and 2), functions and series (Sections 3, 4, and 5), thin sets (Sections 6, 7, 8, 9, and 10). I chose the topics and references according to my own interest, but I decided not to develop what I wrote recently elsewhere [28], so that I was very brief at the end, in particular, on topics I like best. 1. Lebesgue, Baire, and trigonometric series Lebesgue’s measure theory and Baire’s category theorem are both a hundred years old [1,2,43,48]. From the very beginning, they were linked with trigonometric series. Baire was inspired by series of continuous functions: His book “Leçons sur les fonctions discontinues” begins with the example of functions defined as sums of everywhere convergent trigonometric series, the main object of Riemann’s thesis on trigonometric series [3, 53]. The theory of the Lebesgue integral, as we now know it, was first presented in Lebesgue’s book “Leçons su