# Why does the author of this probability book make a certain assumption to state the independence of R.V.'s X and Y?

I have a probability book (for actuarial exam preparation) and the author states as an assumption that "if the probability space is rectangular" in order to show two random variables are independent:

I could not understand why we need this assumption, so I looked at my math stats book, and it appears the author here does not mention the rectangular region assumption:

What is to be made of this?

• It is superfluous. – mathematician Mar 24 '15 at 17:28
• Since the end points are allowed to be infinite, it really says nothing other than that you are dealing with two variables. – copper.hat Mar 24 '15 at 17:36

In the mathematical statistics book, the support of $f(x_1,x_2)$ is the Cartesian product $S_1 \times S_2.$ This is a generalization of the slightly more elementary 'rectangular' space in the actuarial review book, where the two supports are intervals (the usual case in practice).