Vtyjkyuk

I am trying to show that if $epsilon_1$ and $epsilon_2$ are 2 elementary families of sets in $X_1$ and $X_2$ respectively then $$epsilon_1 times epsilon_2=E_1 times E_2 $$ is an elementary family of sets on $X_1 times X_2$.

My book, Folland, defines an elementary family to be a collection
$epsilon$ of subsets of $X$ such that:

1) The empty set is in $epsilon$

2) If $E,F in epsilon$, then $E cap F in epsilon$

3) If $E in epsilon$, then $E^c$ is a finite disjoint union of
members of $epsilon$

For this problem I am trying to show the following:

1) The empty set is in $epsilon_1 times epsilon_2$ is clear since it is in both $epsilon_1$ and $epsilon_2$.

I am not sure about 2) and 3)...

2) I want to show that if $E,F in epsilon_1 times epsilon_2$, then $E cap F in epsilon_1 times epsilon_2$

We know that there exists $E_i,E_m in epsilon_1$ and $E_j,E_n in epsilon_2$ so that $E=E_i times E_j$ and $F=E_m times E_n$. Then, $E cap F=(E_i times E_j) cap (E_m times E_n)$ where both terms in parentheses are in $epsilon_1 times epsilon_2$ by definition. So we have that $E cap F in epsilon_1 times epsilon_2$. I am wondering if this is correct...

3) I want to show that If $E in epsilon_1 times epsilon_2$, then $E^c$ is a finite disjoint union of members of $epsilon_1 times epsilon_2$.

Again, we know that there exists $E_i in epsilon_1$ and $E_j in epsilon_2$ so that $E=E_i times E_j$. I then said that $E^c=(E_i times E_j)^c = (E_i^c times X_1) cup (E_j^c times X_2)$ but I am stuck because $(E_i^c times X_1)$ and $(E_j^c times X_2)$ are not disjoint as far as I understand...

Thank you for any help you can give me.

For 3, let $Einvarepsilon_1$ and $E'invarepsilon_2$. Then
$X_1=E_0cup E_1cupcdotscup E_m$ where this is a disjoint union of
elements of $varepsilon_1$ and $E_0=E$. Likewise
$X_2=E_0'cup E_1'cupcdotscup E_n'$ where this is a disjoint union of
elements of $varepsilon_2$ and $E_0'=E'$.

Then $(Etimes E')^c$ is the disjoint union of the $E_itimes E_j'$
for $(i,j)ne(0,0)$.