Find a family $\mathscr{F}$ such that $\mathscr{F}\subset \mathcal{M}\subset{\Phi_\mathscr{F}}$ with proper subsets.

Question

Find a family $\varnothing\neq\mathscr{F}\subset \mathcal{P}(X)$ where $\Phi_\mathscr{F}$ is the $\sigma-$algebra generated by $\mathscr{F}$, $\mathcal{M}_\mathscr{F}$ the monotone class that contains $\mathscr{F}$ and it's also the smallest one. Show that $$\mathscr{F}\subset \mathcal{M}_\mathscr{F}\subset{\Phi_\mathscr{F}}$$ and find an example where $\subset$ are all proper subsets.

I was able to show $\mathscr{F}\subset \mathcal{M}\subset{\Phi_\mathscr{F}}$but I'm struggling on getting an example. I thought about $$\mathscr{F}=\{\varnothing,\mathbb{Z}\}\cup\{S\subset\mathbb{Z}:|S|<\infty\}$$ and in this case, $\mathscr{F}\subset\mathcal{M}_\mathscr{F}$ strictly but $\mathcal{M}_\mathscr{F}=\Phi_\mathscr{F}$. I also tried with finite sets but in almost all cases I tried equaility of sets holds, and another case where the second $\subset$ was the proper subset.

Answer 1

$X=\mathbb R$ and $\mathcal F= \{(0,1), (0,2), (0,3),\cdots\}$. Here $\mathcal M_{\mathcal F}$ contains $\bigcup_n (0,n+1)=(0,\infty)$ which is not in $\mathcal F$ and $\Phi_{\mathcal F}$ contains $[1,2)$ which is not in $\mathcal M_{\mathcal F}$. In fact, $\mathcal M_{\mathcal F}$ is nothing but $\mathcal F$ together with $(0,\infty)$.

Answer 2

Consider $$\mathscr{F}= \{(-\infty,k] \cap \mathbb{Z}\}_{k \in \mathbb{Z}}\,.$$ For this choice, $\mathcal{M}_\mathscr{F}=\{\varnothing,\mathbb{Z}\}\cup\mathscr{F}$ and $\Phi_\mathscr{F}=\mathcal{P}(\mathbb{Z})$.