This is a comment from Folland when he talks about product measures to justify that the product pre-measure is well defined. The comment is made in the last line of the picture:
My Question: I wonder why such a common refinement exists? It is clear from a picture, but I am struggling to come up with a justification using words. Here I rephrase the statement I wish to prove: Let $\bigcup_{k = 1} ^N (A_k \times B_k)$ be a finite disjoint union of rectangles. If we also have $$ \bigcup_{k = 1} ^N (A_k \times B_k) = \bigcup_{j = 1} ^{\hat{N}} (\hat{A}_j \times \hat{B}_j) $$ with $\{ \hat{A}_j \}_{j = 1} ^{\hat{N}} \subseteq \mathcal{F}$, $\{ \hat{B}_j \}_{j = 1} ^{\hat{N}} \subseteq \mathcal{G}$ and $\{ \hat{A}_j \times \hat{B}_j \}_{j = 1} ^{\hat{N}} \subseteq P(X \times Y)$ be mutually disjoint. Then we can find a finite set of disjoint sets $\{ F_l \}_{l = 1} ^L \in \mathcal{F}$, $\{ G_l \}_{l = 1} ^L \in \mathcal{G}$ with $L \in \mathbf{N}$ such that for all $A_k \times B_k$ and $\hat{A}_j \times \hat{B}_j$, there exists $L_1, L_2 \in \{ 1, \cdots, L \}$ such that $$ A_k \times B_k = \bigcup_{l = 1} ^{L_1} F_l \times G_l $$ and $$ \hat{A}_j \times \hat{B}_j = \bigcup_{l = 1} ^{L_2} F_l \times G_l. $$ Here is my proof so far: We claim if we can find a common refinement of sets in a one dimensional space, we can find a common refinement of sets in any finite dimensional spaces. In particular, given the space $X \times Y$ and sets $\{ A_k \}_{k = 1} ^N \subseteq \mathcal{F}$, $\{ B_k \}_{k = 1} ^N \subseteq \mathcal{G}$, $\{ \hat{A}_k \}_{k = 1} ^N \subseteq \mathcal{F}$, $\{ \hat{B} \}_{k = 1} ^N \subseteq \mathcal{G}$, we may find common refinement of $\{ A_k \}_{k = 1} ^N$ and $\{ \hat{A}_k \}_{k = 1} ^{\hat{N}}$ to be $\{ F_l \}_{l = 1} ^{L_1}$ and the common refinement of $\{ B_k \}_{k = 1} ^N$ and $\{ \hat{B}_k \}_{k = 1} ^{\hat{N}}$ to be $\{ G_h \}_{h = 1} ^{L_2}$. Then we for all $k \in \mathbf{N}$, $$ A_k \times B_k = \bigcup_{l = 1} ^{L_1} F_l \times \bigcup_{h = 1} ^{L_2} G_l = \bigcup_{l = 1} ^{L_1} \bigcup_{h = 1} ^{L_2} F_l \times G_h. $$ Then I am stuck, noticing my claim might not be true? I am not sure how to fix this as the index of $F_l$ and $G_h$ is not the same. Is there a easier and better way to do this?