Suppose we have sigma finite measure spaces $(X,\mathcal{M},\mu)$, $(Y,\mathcal{M},\nu)$. Let $X_j$, $Y_j$ be the exhausting sequences for $X,Y$. We know that given two measure spaces, we can generate the product measure space $(X_j \times Y_j, (\mathcal{M} \otimes \mathcal{N}), (\mu\times\nu)$. As $(X_j \times Y_j) \subset \mathcal{M} \otimes \mathcal{N}$, we can deduce that the restriction, $(X_j \times Y_j, (\mathcal{M} \otimes \mathcal{N}) \cap (X_j \times Y_j), (\mu\times\nu)_{(X_j \times Y_j)}(\cdot))$, is a measure space. Similarly, we deal with restricted measure spaces $(X_j, \mathcal{M} \cap X_j, \mu_{X_j}(\cdot) = \mu(\cdot \cap X_j))$, $(Y_j, \mathcal{N} \cap Y_j, \nu_{Y_j}(\cdot) = \mu(\cdot \cap Y_j))$ where $\mu_{X_j}$ is the restriction of $\mu$ to $X_j$, and same for $\nu$.
Prove that the measure space generated from these two measure spaces, $(X_i\times Y_i,(\mathcal{M} \cap X_j) \otimes (\mathcal{N} \cap Y_j), (\mu_{X_j} \times \nu_{Y_j}))$, is equal to $(X_j \times Y_j, (\mathcal{M} \otimes \mathcal{N}) \cap (X_j \times Y_j), (\mu\times\nu)_{(X_j \times Y_j)}(\cdot))$. I feel like this should be easy, but I am having difficulty. Does anyone know how to prove this? Here is my attempt:
(Attempted proof) I claim $(\mathcal{M} \cap X_j) \otimes (\mathcal{N} \cap Y_j) \subset \mathcal{M} \otimes \mathcal{N} \cap (X_j \times Y_j)$. Due to results we proved for product measures, we know that $(\mathcal{M} \cap X_j) \otimes (\mathcal{N} \cap Y_j)$ is generated by sets of form $A \times B$, where $A \in \mathcal{M} \cap X_j$ and $B \in \mathcal{N} \cap Y_j$. Thus, $\exists A_1 \in \mathcal{M}$, $\exists B_1 \in \mathcal{N}$ such that $A = A_1 \cap X_j$, $B = B_1 \cap Y_j$. Note that $A_1 \cap X_j \times B_1 \cap Y_j = (A_1 \times B_1) \cap (X_j \times Y_j)$. This is an element of $\mathcal{M} \otimes \mathcal{N} \cap (X_j \times Y_j)$.
Secondly, I claim that $(\mathcal{M} \cap X_j) \otimes (\mathcal{N} \cap Y_j) \supset \mathcal{M} \otimes \mathcal{N} \cap (X_j \times Y_j)$. As the basis for $\mathcal{M} \otimes \mathcal{N}$ are sets of form $A_1 \times B_1$, where $A_1 \in \mathcal{M}$ and $B_1 \in \mathcal{N}$ ... Actually, it is not so obvious that the basis for $\mathcal{M} \otimes \mathcal{N} \cap (X_j \times Y_j)$ are sets of form $(A \times B) \cap (X_j \times Y_j)$ where $A \in \mathcal{M}$, $B \in \mathcal{N}$. Anyhow, I cannot think of how to prove this. Help?