Weighted symmetric functions (Q2640716)

In an earlier paper [Real Anal. Exch. 14, No.2, 429-439 (1989; Zbl 0678.26003)] the author studied the classes of weighted symmetric functions as a generalization of symmetric and symmetrically continuous functions. In the present paper, the author generalizes some results obtained by \textit{H. Auerbach} [Fundam. Math. 8, 49-55 (1926)], \textit{M. Mazurkiewicz} [Fundam. Math. 11, 145-147 (1928)] and \textit{C. J. Neugebauer} [Duke Math. J. 31, 23-31 (1964; Zbl 0125.307)] concerning measurable symmetric functions to functions symmetric with respect to an even weight system. Crucial in the proof of the main theorems is the following proposition: If 0 is a point of density for a measurable set E, m is a fixed number, \(0<m\leq 1\), and \(\{a_ i,b_ i\},\quad i=1,2,...,n,\) is a set of real numbers such that \(b_ i=0,\quad i=1,2,...,n,\) then for all positive u sufficiently small, there is a v in \([mu,(m+1)u]\) such that \(a_ iu+b_ iv\in E,\quad i=1,2,...,n.\) This proposition generalizes the results of \textit{A. Zygmund} [Fundam. Math. 26, 1-43 (1936; Zbl 0014.11102)] and \textit{J. M. Ash} [Trans. Am. Math. Soc. 126, 181-199 (1967; Zbl 0164.359)], the proof being completely similar to that of lemma 1 from the above mentioned Ash's paper.