EDIT: I retract the following. The problem with it is that Coscott is arguing that "something in the real world that you can throw a dart at" implies "measurable" and he does this by arguing that all sets which are "something in the real world that you can throw a dart at" have a certain property which implies measurability. My "counterexamples" are measurable sets which fail to have this property, but this is the opposite of what I would need to disprove him. I'd need to find a set with this property that isn't measurable. In fact, I don't think there is such a set; I think Coscott is right.
The sets with this property (that you can tell whether your number is in or out after only finitely many dice rolls) are the open sets, not the measurable sets. For example, the set [0,pi-3] is measurable but not open. If the die comes up (1,4,1,5,9,...) then you'll never know if your number is in or out until you have all the digits. For an even worse example take the rational numbers: they're measurable (measure zero) but any finite decimal expansion could be leading to a rational or an irrational.
Yeah, but I can't explain explain that without analysis not appropriate for a less wrong post. I remember that the probability class I took in undergrad dodged the measure theory questions by defining probabilities on open sets, which actually works for most reasonable questions. I think such a simplification is appropriate, but I should have had a disclaimer.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.