Approximation and extension of random functions (Q1263147)

Stochastic versions of extension theorems of Tietze and \textit{J. Dugundji} [Pac. J. Math., 1, 353-367 (1951; Zbl 0043.381)] are obtained, as well as an existence theorem for partitions of unity by random continuous functions. A form of the classical approximation theorem of \textit{S. N. Mergelyan} [Uspechi Mat. Nauk 7, No.2(48), 31-122 (1952; Zbl 0049.327); English translation in Amer. Math. Soc., Translat. No.101, 99 p. (1954)] valid for random holomorphic functions on random compact sets is presented. A similar approach yields versions of the approximation theorems of \textit{C. Runge} [Acta Math. 6, 229-244 (1885)], \textit{N. U. Arakelyan} [Izv. Akad. Nauk SSSR, Ser. Mat. 28, 1187-1206 (1964; Zbl 0143.296)], and \textit{A. G. Vitushkin} [Sov. Math., Dokl. 7, 1622-1625 (1967; Zbl 0162.097)].