Reference: Introductory Real Analysis, Kolomogorov and Fomin, Dover Publications, Translated and edited by Richard A. Silverman:
Let X be an uncountable set, and be the ring consisting of all finite subsets of X and their complements. Is a -ring also?
Are open intervals Borel sets ?
Let be a function defined on a set M and taking values in a set N. Let be a system of subsets of M, and let denote the system of all images of sets . Moreover, let be a system of subsets of N, and let denote the system of all preimages of of sets . Prove that
(i) If is a ring, so is
(ii) If is an algebra, so is
(iii) If is a borel algebra, then so is
Which of these assertions remain true if os replaced by and by f?