Exercises based on system of sets

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 \mathscr{R} be the ring consisting of all finite subsets of X and their complements. Is \mathscr{R} a \sigma -ring also?

Problem 2:

Are open intervals Borel sets ?

Problem 3:

Let y=f(x) be a function defined on a set M and taking values in a set N. Let \mathscr{M} be a system of subsets of M, and let f(\mathscr{M}) denote the system of all images f(A) of sets A \in \mathscr{M}. Moreover, let \mathscr{N} be a system of subsets of N, and let f^{-1}(\mathscr{N}) denote the system of all preimages of f^{-1}(B) of sets B \in \mathscr{N}. Prove that

(i) If \mathscr{N} is a ring, so is f^{-1}(\mathscr{N})

(ii) If \mathscr{N} is an algebra, so is f^{-1}()\mathscr{N}

(iii) If \mathscr{N} is a borel algebra, then so is f^{-1}(\mathscr{N})

(iv) \mathscr{R}(f^{-1}(\mathscr{N}))=f^{-1}(\mathscr{R}(\mathscr{N}))

(v) \mathscr{R}(f^{-1}(\mathscr{N}))=f^{-1}(\mathscr{R}(\mathscr{N})).

Which of these assertions remain true if \mathscr{N} os replaced by \mathscr{M} and f^{-1} by f?

Regards,

Nalin Pithwa

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.