I know the following defintion for a Product-Sigma-Algebra:
Let be $ (\Omega_1,\mathcal{A}_1,\mu_1) $ and $ (\Omega_2,\mathcal{A}_2,\mu_2) $ two measure spaces. The Sigma-Algebra over $ \Omega_1\times \Omega_2 $ generated by the sets of the form $ A_1\times A_2 , A_i\in \mathcal{A}_i, i=1,2 $ is called Product-Sigma-Alegbra of $ \mathcal{A}_1 $ and $ \mathcal{A}_2 $ and is named by $ \mathcal{A}_1\otimes \mathcal{A}_2 $ such that $ \mathcal{A}_1\otimes \mathcal{A}_2=\sigma\left(\{A_1 \times A_2:\ (A_1,A_2)\in \mathcal{A}_1\times \mathcal{A}_2\}\right) $.
Why does contain $ \mathcal{A}_1\otimes \mathcal{A}_2 $ the set $ \mathcal{A}_1\times \mathcal{A}_2 $ and in which relation? Is it like "$\in$" or "$\subseteq$"?
I tried to comprehend it by an example:
$ \Omega_1=\{7\}, \mathcal{A}_1=\{\emptyset, \{7\}\}$ and $\Omega_2=\{1,2\},\mathcal{A}_2=\{\emptyset, \{1\},\{2\},\{1,2\}\} $ and $ \Omega_1\times \Omega_2=\{(7,1),(7,2)\} $.
Now I calculate $$\mathcal{A}_1\times \mathcal{A}_2=\{(\emptyset,\emptyset),(\emptyset,\{1\}),(\emptyset,\{2\}),(\emptyset,\{1,2\}),(\{7\},\emptyset),(\{7\},\{1\}),(\{7\},\{2\}),(\{7\},\{1,2\})\}$$
and $$ \begin{aligned}&\{A_1 \times A_2:\ (A_1,A_2)\in \mathcal{A}_1\times \mathcal{A}_2\}\\[10pt]=&\{\emptyset\times \emptyset,\emptyset\times \{1\},\emptyset\times \{2\},\emptyset\times \{1,2\},\{7\}\times \emptyset, \{7\}\times \{1\}, \{7\}\times \{2\}, \{7\}\times \{1,2\}\}\\[10pt]=&\{\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\{(7,1)\},\{(7,2)\},\{(7,1),(7,2)\}\}\\[10pt]=&\{\emptyset,\{(7,1)\},\{(7,2)\},\{(7,1),(7,2)\}\}. \end{aligned} $$
Finally I get $$ \mathcal{A}_1\otimes \mathcal{A}_2=\sigma\left(\{A_1 \times A_2:\ (A_1,A_2)\in \mathcal{A}_1\times \mathcal{A}_2\}\right)=\{\emptyset,\{(7,1)\},\{(7,2)\},\{(7,1),(7,2)\}\} $$ but the elements in the set $ \mathcal{A}_1\times \mathcal{A}_2 $ are different. What went wrong?
Best Answer
Nothing went wrong.
Note that $\mathcal{A}_{1}\times\mathcal{A}_{2}\subseteq\mathcal P(\Omega_1)\times\mathcal P(\Omega_2)$ and $\mathcal{A}_{1}\otimes\mathcal{A}_{2}\subseteq\mathcal P(\Omega_1\times\Omega_2)$.
We not expected to have $\mathcal{A}_{1}\times\mathcal{A}_{2}\subseteq\mathcal{A}_{1}\otimes\mathcal{A}_{2}$.
What we must have is: $$\left\{ A_{1}\times A_{2}\mid A_{1}\in\mathcal{A}_{1},A_{2}\in\mathcal{A}_{2}\right\} \subseteq\mathcal{A}_{1}\otimes\mathcal{A}_{2}$$or equivalently:$$\{A_1\times A_2\mid (A_1,A_2)\in \mathcal A_1\times\mathcal A_2\}\subseteq\mathcal{A}_{1}\otimes\mathcal{A}_{2}$$
Unfortunately the set on LHS is sometimes denoted as $\mathcal{A}_{1}\times\mathcal{A}_{2}$ which can cause confusion.