In Freidman's Foundations of Modern Analysis, ch. 2.15 Product of Measures, he defines:
- The Cartesian product $X \times Y$ as all the ordered pairs $(x,y)$, where $x \in X$ and $y \in Y$.
- Rectangles as any subset of $X \times Y$ of the form $A \times B$, where $A \subset X$ and $B \subset Y$
- The $\sigma$-algebra $\mathcal{A} \times \mathcal{B}$ as the $\sigma$-algebra generated by all the rectangles $A \times B$ with $A \in \mathcal{A}$ and $B \in \mathcal{B}$, where $\mathcal{A}$ is a $\sigma$-algebra of subsets of $X$ and $\mathcal{B}$ is a $\sigma$-algebra of subsets of $Y$.
I have no problem with the definitions so far, but then he writes (p. 79):
- "A rectangle $A \times B$ with $A \subset \mathcal{A}$, $B \in \mathcal{B}$ will be called a rectangle of $\mathcal{A} \times \mathcal{B}$."
I don't understand why we would mix a subset of a $\sigma$-algebra $A \subset \mathcal{A}$ with a member of a of a $\sigma$-algebra $B \in \mathcal{B}$. I considered a simple example:
$X = \{1, 2, 3\}$, $Y = \{-1, 4\}$, $\mathcal{A}= \{\emptyset, \{1\}, \{2,3\}, X\}$ and $\mathcal{B}= \{\emptyset, \{-1\}, \{4\}, Y\}$
We would then for example have that $\{(2,4), (3,4)\} \in \mathcal{A} \times \mathcal{B}$. "A rectangle $A \times B$ with $A \subset \mathcal{A}$, $B \in \mathcal{B}$" would then for example correspond to $\{(\emptyset, -1), (\{1\}, -1)\}$ which doesn't seem right (also the rest of the text becomes unclear to me). Could someone please help me make sense of the forth bullet point above?