I recently learned about algebras, $\sigma$-algebras, and monotone classes. The algebras and $\sigma$-algebras I feel are quite clear, but I think I may be a bit confused about certain aspects of monotone classes. The following is the definition of monotone classes given:
A non-empty family $\Phi \subset \mathcal{P}(X)$ which is closed under countable increasing unions and countable decreasing intersections is called a monotone class in X, i.e. $\Phi$ is a monotone class if
$E_1\subset E_2\subset ...$ where each $E_j\in\Phi$ implies that $\bigcup_{j=1}^{\infty} E_j \in \Phi$;
$D_1\supset D_2\supset ...$ where each $D_j\in\Phi$ implies that $\bigcap_{j=1}^{\infty} D_j \in \Phi$.
Now, suppose that $X=\left\{1,2,3\right\}$, and let $\Phi_1=\left\{\left\{1\right\},\left\{1,2\right\}\right\}$, $\Phi_2=\left\{\emptyset, X\right\}$. Then $\Phi = \Phi_1 \cap \Phi_2=\emptyset$, so $\Phi$ is empty, and therefore, not a monotone class.
But this confuses me a bit, as I don't see clearly from this definition why either of $\Phi_1$ and $\Phi_2$ shouldn't be a monotone class.
My guess is that $\Phi_1$ is not a monotone class and $\Phi_2$ is a monotone class, since $X\in\Phi_2$, and I've read that $X$ is always an element of any monotone class in $X$. But I don't see why that is. What am I not seeing in the definition?
Both $\Phi_1$ and $\Phi_2$ are monotone classes.
Pick an infinite sequence $A_n$ of elements of $\Phi_1$. Then for every $n$ either $A_n=\{1\}$ or $A_n = \{1,2\}$, so the infinite intersection and union is going to be either $\{1,2\}$ or $\{1\}$, both of which belong to $Phi_1$.
A similar argument can be made regarding $\Phi_2$.
Thus we see that it is not necessary to include $X$ in order to be a monotone class.