monotone class theorem failure for a class of subsets that is not a field

Question

Show the monotone class theorem fails if $F_{0}$ is not assumed to be a field.

Monotone class theorem: Let $F_{0}$ be a field of subsets of $\Omega$, and $C$ a class of subsets of $\Omega$ that is monotone. If $C \supset F_{0}$ , then $C \supset \sigma (F_{0})$.

Answer 1

Consider $\Omega=\Bbb{R}$, $C=F_0=\{\Bbb{R},\varnothing\}\cup\{[n,\infty):n\in\Bbb{Z}\}$. Obviously, $C$ is monotone. But $\sigma(F_0)$ contains $(-\infty,0)$, which is not contained in $C$.