Notes on the inaugural lecture delivered by Frederic Riesz in 1925 as rector of Szeged University
This is a slightly modified version of a paper written in Hungarian [Mat. Lapok 9, 232-259 (1959; Zbl 0086.047)]. In his still inspiring inaugural lecture as rector of Szeged University in 1925, F. Riesz explained how an argument like the one used in the proof of the Bolzano-Weierstrass- theorem often enables one to draw a general conclusion from an elementary insight. His lecture culminated in the following theorem: ``A family of circular discs of fixed radius \(\rho\) lies in a bounded planar domain of area T. Then the length of the envelope of the union of the family cannot exceed 2T/\(\rho\) ''. Riesz observed that the theorem is obvious in case the family in question is finite, and then applied the following principle: ``A sequence of curves, each of length \(\leq z\), cannot have a limit curve of length \(>z.''\) The author discusses the validity of this principle. He remarks that the usual notion of length cannot be applied always, and proposes to use the outer linear measure of Carathéodory. He then shows, by giving an example, that the principle enunciated by Riesz is false. However, the principle is valid if one imposes the condition that the discs have all the same radius. Therefore, Riesz's theorem is true, although the proof given by Riesz himself is incorrect. The author establishes the following generalization of the improved principle (we do not quote literally): ``Let w be a positive and z a nonnegative real number. Let \({\mathcal F}\) be a family of subsets of the plane fulfilling the following conditions: (i) The components of the members \({\mathcal F}\) are of diameter \(\geq w\). (ii) Let \({\mathcal F}'\) be a finite subfamily of \({\mathcal F}\). Then the outer linear measure of the boundary of the union of \({\mathcal F}'\) is not greater than z. Then the outer linear measure of the boundary of the union of \({\mathcal F}\) itself is not greater than z.'' The proof takes seventeen steps and twenty pages.
scientific article; zbMATH DE number 4055554
scientific article