Say $\phi\in\Phi$ and $r\in\mathbb R$ where $\Phi$ is an infinite set (possibly uncountable). $A(r,\phi),B(r,\phi)$ are sets (actually, events). I want to write
$$
\bigcap_{r>0}\bigcup_{\phi\in\Phi}(A(r,\phi)\cap B(r,\phi))=\left(\bigcap_{r>0}\bigcup_{\phi\in\Phi}A(r,\phi)\right)\cap\left(\bigcap_{r>0}\bigcup_{\phi\in\Phi}B(r,\phi)\right).
$$
But I doubt if this is justified (I'm not very familiar with set theory)… Since I may oversimplified the conditions, if the above holds conditionally it would also be OK.
I would be extremely appreciative of any assistance!
Best Answer
$$x\in\bigcap_{r>0}\bigcup_{\phi\in\Phi}(A(r,\phi)\cap B(r,\phi))\iff\forall r>0\;\exists\phi\in\Phi\;x\in A(r,\phi)\land x\in B(r,\phi)$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\iff x\in\big(\bigcap_{r>0}\bigcup_{\phi\in\Phi}A(r,\phi)\big)\cap\big(\bigcap_{r>0}\bigcup_{\phi\in\Phi}B(r,\phi)\big)$$