I understand the concept of independence to be dichotomous- events are either independent or dependent. And while there are infinitely many ways for events to be dependent and only one way to be independent, independent events are still likely and 'common'.
However, if rho is a continuous measure of (linear independence), it is thus that P(rho=0) is 0.
So if the probability of rho being exactly zero is zero, then does it follow that the probability of two random variables being independent is also zero?