Logan Wyatt

2022-06-30

Let $X=Y$ be uncountable and define $\mathcal{A},\mathcal{B}$ to be the countable-cocountable $\sigma$-algebras on $X,Y$ respectively. Let $C=\sigma \left(A×B\right)$. Prove that for each subset $E\in C$, there exist $A,B$ in $\mathcal{A},\mathcal{B}$ countable such that either $E\subseteq \left(A×Y\right)\cup \left(X×B\right)$ or ${E}^{c}\subseteq \left(A×Y\right)\cup \left(X×B\right)$.
My idea is to define $D$ as the collection of all such subsets of $C$, prove that $D$ contains ${A}_{1}×B1$ for each ${A}_{1}\in \mathcal{A},{B}_{1}\in \mathcal{B}$ and then show that $D$ is a $\sigma$-algebra.
I am able to prove that $D$ is a $\sigma$ algebra, and I can show that in the case ${A}_{1},{B}_{1}$ are either both countable or cocountable, ${A}_{1}×{B}_{1}$ is in $D$. How do I do it for the case that exactly one of them is cocountable?
Something that may be useful is
$\left({A}_{1}×{B}_{1}{\right)}^{c}=\left({A}_{1}^{c}×Y\right)\cup \left(X×{B}_{1}^{c}\right)\cup \left({A}_{1}^{c}×{B}_{1}^{c}\right)$

Bruno Dixon

Expert

Suppose ${A}_{1}$ is countable and ${B}_{1}$ is cocountable. Then it follows that
${A}_{1}×{B}_{1}\subseteq {A}_{1}×Y\subseteq \left({A}_{1}×Y\right)\cup \left(X×B\right)$
where $B\in \mathcal{B}$ is an arbitrary countable set (e.g., $B=\mathrm{\varnothing }$). This shows that ${A}_{1}×{B}_{1}\in D$.

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