I have a question about a problem in Probability & Measure Theory by Ash and Doléans-Dade. It concerns the following theorem:
1.3.9 Monotone Class Theorem. Let $\mathscr F_0$ be a field of subsets of $\Omega$, and $\mathscr C$ a class of subsets of $\Omega$ that is monotone (if $A_n \in \mathscr C$ and $A_n \uparrow A$ or $A_n \downarrow A$, then $A \in \mathscr C$). If $\mathscr C \supset \mathscr F_0$, then $\mathscr C \supset \sigma(\mathscr F_0)$, the minimal $\sigma$-field over $\mathscr F_0$.
Here is the problem:
Show that the monotone class theorem (1.3.9) fails if $\mathscr F_0$ is not assumed to be a field.
To me there are two ways one could reasonably interpret this problem:
Show that there exists a set $\Omega$ with a (nonfield) class $\mathscr F_0$ of subsets and a monotone class $\mathscr C$ of subsets such that $\mathscr C \supset \mathscr F_0$ and $\mathscr C \not\supset \sigma(\mathscr F_0)$.
Given a set $\Omega$ and a (nonfield) class $\mathscr F_0$ of subsets of $\Omega$, show that there exists a monotone class $\mathscr C$ of subsets of $\Omega$ such that $\mathscr C \supset \mathscr F_0$ and $\mathscr C \not\supset \sigma(\mathscr F_0)$.
Interpretation 1 is easy to solve, for example by taking $\Omega$ to be any set and $\mathscr F_0 = \mathscr C = \emptyset$, or as they did in this related question.
But maybe interpretation 2 is the intended one? This is a much stronger statement, and seems much harder to prove, if it is even true. So before I spend more time trying to prove it I would like to ask the community if the conclusion of is even true? If it is, do you have any advice on how one might approach solving it?