I'm studying measure theory and I cannot wrap my head around this:
$A_{\omega_1} = \bigcup_{n=1}^\infty (A_n)_{\omega_1}$
The definition is $A_{\omega_1} = \begin{cases} A_2 & \omega_1 \in A_1 \\ \emptyset & otherwise \end{cases}$ and we have $A = \bigcup_{n=1}^\infty A_n$ and for any $A_n$ we have $A_n = A_{n,1} \times A_{n,2}$
So what I'm thinking is, $\exists n$ such that $\omega_1 \in A_{n,1}$. Let all such $n$ form index set $I$ and therefore $\bigcup_{n=1}^\infty (A_n)_{\omega_1} = \bigcup_{n\in I} (A_n)_{\omega_1} = \bigcup_{n\in I} A_{n,2}$
But if $I$ is not empty, then $A_{\omega_1} = \left(\bigcup_{n=1}^\infty A_n\right)_{\omega_1} = \bigcup_{n=1}^\infty A_{n,2}$
So why $A_{\omega_1} = \bigcup_{n=1}^\infty (A_n)_{\omega_1}$ instead of $A_{\omega_1} \supseteq \bigcup_{n=1}^\infty (A_n)_{\omega_1}$
?
EDIT: I think I figured it out.
My problem was assuming that if you had $A_n = B_n \times C_n$ then $$\bigcup_n A_n = \bigcup_n B_n \times \bigcup_n C_n$$ but actually $$\bigcup_n A_n \subseteq \bigcup_n B_n \times \bigcup_n C_n$$
So in light of this, the definition $A_{\omega_1} = \begin{cases} A_2 & \omega_1 \in A_1 \\ \emptyset & otherwise \end{cases}$ does not make sense when applied like this $\left(\bigcup_{n=1}^\infty A_n\right)_{\omega_1}$, that is not well defined. So therefore $A_{\omega_1} = \bigcup_{n=1}^\infty (A_n)_{\omega_1}$ is just the definition. I think the book was just not clear about that.