Why need the finiteness of $\mu(A)$ and $\mu(B)$ to define measurable rectangle $A\times B$ in Royden?

In this errata of Real Analysis by Royden and PM Fitzpatrick, it corrects the definition of measurable rectangles to be

$A\times B$ if $A\in\mathcal{A}, B\in\mathcal{B}$, and $\mu(A)$ and $\mu(B)$ are finite

at the right beginning of chapter 20. This correction makes this measurable rectangle definition different from others which usually don't require the finiteness of $\mu(A)$ and $\mu(B)$. The problem is I don't see the reason why we need this finiteness in its following argument and proof of Fubini theorem.

Update on 20th, Feb, 2017

I guess the reason why the book defines measurable rectangle this way is to avoid $0\cdot\infty$ when defining outer measure by those rectangles.