Example of a field that is not a monotone class.
I tried to come up with a field for which the limit would be a singleton but the finite union or complements or intersections wouldn't be that. Define $$B_n=\left\{x\in\mathbb{R}:x<\frac{1}{n}\right\}$$ Clearly, $B_1\supset B_2\supset B_3...$ and $\cap_{i=1}^nB_i=B_n$ but, $\cap_{i=1}^\infty B_i=\{0\}$, which cannot be expressed as a finite union of any $B_n$'s or their complements. Therefore, the minimal field generated by $\{B_n\}_{n\in\mathbb{N}}$ is not a monotone class.
Is the above example valid? Is there an "easier" example (in the sense that it is easier to prove)?
Take the field of all finite or cofinite subsets of a given infinite set.