Gretchen Schwartz

2022-07-01

I am reading one book and the author presents what he calls a distributivity formula:
$\bigcup _{j\in J}\bigcap _{i\in {I}_{j}}{F}_{i}^{j}=\bigcap _{\left\{{i}_{j}\right\}\in K}\bigcup _{j\in J}{F}_{{i}_{j}}^{j}$
where $K={\mathrm{\Pi }}_{j\in J}{I}_{j}$ (the set of all sequences {${i}_{j},j\in J$}).
My question is regarding the notation. The LHS is quite clear to me. However, I'm now sure what are actually the members of K. Say $J=\left\{1,2,\dots \right\}$ and ${I}_{j}=\left\{1,2,\dots \right\}$. What is actually ${i}_{j}$ here?

Marisol Morton

Expert

On the RHS, we are finding the intersection of $\bigcup _{j\in J}{F}_{{i}_{j}}^{j}$. Here, the ${i}_{j}$ is the values for which we are intersecting; it says $\left\{{i}_{j}\right\}\in K$. If $J=\left\{1,2,3,\cdots \right\}$, then $K={I}_{1}×{I}_{2}×{I}_{3}×\cdots$