I should have been more accurate and said "If the random number that you'll eventually get does in fact lie in the set, then you'll find out about this fact after a finite number of rolls."

This really does define open sets, since for any point in an open set there's an open ball of radius epsilon about it which is in the set, and then the interval [x, x+1/10^n] has to be in that ball once 1/10^n < epsilon/2.

EDIT: (and the converse also holds, I think, but it requires some painfully careful thinking because of the non-uniqueness of decimal expansions)

I think a more exact representation of what Coscott actually said is the following property: "We almost always only have to roll the die finitely many times to determine whether the point is in or out."

This still doesn't specify measurable sets (because of the counterexample given by the rationals). I think the type of set that this defines is "Sets with boundary of measure zero" where the boundary is the closure minus the interior. Note that the rationals in [0,1) have boundary everywhere (i.e. boundary of measure 1).