There are non-measurable sets (unless you discard the axiom of choice, but then you’ll run into some significant problems.) Indeed, the existence of non-measurable sets is the reason for so much of the measure-theoretic formalism.
This depends on the space.
It’s at least true for real-valued intervals with continuous measures, of course, but I think you’re never going to ask for the measure of a non-measurable set in real-world applications, precisely because they require the axiom of choice to construct (at least for the real numbers, and I’d assume, by extension, any subset of any Rn), and no natural set you’ll be interested in that comes up in an application will require the axiom of choice (more than dependant choice) to construct. I don’t think the existence of non-measurable sets is viewed as a serious issue for applications.
It is not true in a countable measure space (or, at least, you could always extend the measure to get this to hold), since assuming each singleton (like {x},x∈X) is measurable, every union of countably many singletons is measurable, and hence every subset is measurable (A=∪x∈A{x} is a countable union of singletons, A⊆X, X countable) . In particular, if you’re just interested in the number of future people, assuming there are at most countably infinitely many (so setting aside the many-worlds interpretation of quantum mechanics for now), then your space is just the set of non-negative integers, which is countable.
using infinite sets (which clearly one would have to do if reasoning about all possible futures)
You could group outcomes to represent them with finite sets. Bayesians get to choose the measure spaces/propositions they’re interested in. But again, I don’t think dealing with infinite sets is so bad in applications.
This depends on the space.
It’s at least true for real-valued intervals with continuous measures, of course, but I think you’re never going to ask for the measure of a non-measurable set in real-world applications, precisely because they require the axiom of choice to construct (at least for the real numbers, and I’d assume, by extension, any subset of any Rn), and no natural set you’ll be interested in that comes up in an application will require the axiom of choice (more than dependant choice) to construct. I don’t think the existence of non-measurable sets is viewed as a serious issue for applications.
It is not true in a countable measure space (or, at least, you could always extend the measure to get this to hold), since assuming each singleton (like {x},x∈X) is measurable, every union of countably many singletons is measurable, and hence every subset is measurable (A=∪x∈A{x} is a countable union of singletons, A⊆X, X countable) . In particular, if you’re just interested in the number of future people, assuming there are at most countably infinitely many (so setting aside the many-worlds interpretation of quantum mechanics for now), then your space is just the set of non-negative integers, which is countable.
You could group outcomes to represent them with finite sets. Bayesians get to choose the measure spaces/propositions they’re interested in. But again, I don’t think dealing with infinite sets is so bad in applications.