Exercises based on system of sets

Posted on January 3, 2020January 3, 2020 by Nalin Pithwa

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

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