I’ve been thinking about this a bit more. My current thinking is basically what Coscott said:
We only care about probabilities if we can be forced to make a bet.
In order for it to be possible to decide who won the bet, we need that (almost always) a measurement to some finite accuracy will suffice to determine whether the dart is in or out of the set.
Thus the set has a boundary of measure zero.
Thus the set is measurable.
What we have shown is that in any bet we’re actually faced with, the sets involved will be measurable.
(The steps from 2 to 3 and 3 to 4 are left as exercises. (I think you need Lebesgue measurable sets rather than just Borel measurable ones))
Note that the converse fails: I believe you can’t make a bet on whether or not the dart fell on a rational number, even though the rationals are measurable.
Here’s a variant which is slightly different, and perhaps stronger since it also allows some operations with “infinite accuracy”.
In order to decide who won the bet, we need a referee. A natural choice is to say that the referee is a Blum-Shub-Smale machine, i.e. a program that gets a single real number x∈[0,1] as input, and whose operations are: loading real number constants; (exact) addition, substraction, multiplication and division; and branching on whether a≤b (exactly).
Say you win if the machine accepts x in a finite number of steps. Now, I think it’s always the case that the set of numbers which are accepted after n steps is a finite union of (closed or open) intervals. So then the set of numbers that get accepted after any finite number of steps is a countable union of finite unions of intervals, hence Borel.
I’ve been thinking about this a bit more. My current thinking is basically what Coscott said:
We only care about probabilities if we can be forced to make a bet.
In order for it to be possible to decide who won the bet, we need that (almost always) a measurement to some finite accuracy will suffice to determine whether the dart is in or out of the set.
Thus the set has a boundary of measure zero.
Thus the set is measurable.
What we have shown is that in any bet we’re actually faced with, the sets involved will be measurable.
(The steps from 2 to 3 and 3 to 4 are left as exercises. (I think you need Lebesgue measurable sets rather than just Borel measurable ones))
Note that the converse fails: I believe you can’t make a bet on whether or not the dart fell on a rational number, even though the rationals are measurable.
Here’s a variant which is slightly different, and perhaps stronger since it also allows some operations with “infinite accuracy”.
In order to decide who won the bet, we need a referee. A natural choice is to say that the referee is a Blum-Shub-Smale machine, i.e. a program that gets a single real number x∈[0,1] as input, and whose operations are: loading real number constants; (exact) addition, substraction, multiplication and division; and branching on whether a≤b (exactly).
Say you win if the machine accepts x in a finite number of steps. Now, I think it’s always the case that the set of numbers which are accepted after n steps is a finite union of (closed or open) intervals. So then the set of numbers that get accepted after any finite number of steps is a countable union of finite unions of intervals, hence Borel.