Reference: Introductory Real Analysis, Kolomogorov and Fomin, Dover Publications, Translated and edited by Richard A. Silverman:
Problem 1:
Let X be an uncountable set, and be the ring consisting of all finite subsets of X and their complements. Is
a
-ring also?
Problem 2:
Are open intervals Borel sets ?
Problem 3:
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
(iv)
(v) .
Which of these assertions remain true if os replaced by
and
by f?
Regards,
Nalin Pithwa