I observe that I probably miscommunicated. I think multiple people took me to be arguing for a space of lotteries with finite support. That is NOT what I meant. That is sufficient, but I meant something more general when I said “lotteries closed under finite mixtures” I did not mean there only finitely many atomic worlds in the lottery. I only meant that there is a space of lotteries, some of which maybe have infinite support if you want to think about atomic worlds, and for any finite set of lotteries, you can take a finite mixture of those lotteries to get a new lottery in the space. The space of lotteries has to be closed under finite mixtures for VNM to make sense, but the emphasis is on the fact that it is not closed under all possible countable mixtures, not that the mixtures have finite support.
I observe that I probably miscommunicated. I think multiple people took me to be arguing for a space of lotteries with finite support. That is NOT what I meant. That is sufficient, but I meant something more general when I said “lotteries closed under finite mixtures” I did not mean there only finitely many atomic worlds in the lottery. I only meant that there is a space of lotteries, some of which maybe have infinite support if you want to think about atomic worlds, and for any finite set of lotteries, you can take a finite mixture of those lotteries to get a new lottery in the space. The space of lotteries has to be closed under finite mixtures for VNM to make sense, but the emphasis is on the fact that it is not closed under all possible countable mixtures, not that the mixtures have finite support.