There’s a step in one of your proofs that I don’t quite follow. It seems to depend on some sort of continuity axiom, perhaps implicit in the notion of infinite lotteries. It’s where you go from
X2∞≥127X1+1481X1+64243X1+128729X1+...
to
X2∞≥X1
I can see that the sum of the infinite sequence of coefficients is 1, but as elsewhere in the post you’ve avoided ever computing infinite sums, as that sort of computation is not a thing in this setup, I’m not seeing the justification here.
ETA: I think I have dissolved my confusion. Expressions of the form p0X0+p1X1+p2X2+…, finite or infinite, denote probability distributions, and therefore e.g.12X+12X denotes the same distribution as X. The infinite sequence that I cited is by definition the same thing as X1.
There’s a step in one of your proofs that I don’t quite follow. It seems to depend on some sort of continuity axiom, perhaps implicit in the notion of infinite lotteries. It’s where you go from
X2∞≥127X1+1481X1+64243X1+128729X1+...
to
X2∞≥X1
I can see that the sum of the infinite sequence of coefficients is 1, but as elsewhere in the post you’ve avoided ever computing infinite sums, as that sort of computation is not a thing in this setup, I’m not seeing the justification here.
ETA: I think I have dissolved my confusion. Expressions of the form p0X0+p1X1+p2X2+…, finite or infinite, denote probability distributions, and therefore e.g.12X+12X denotes the same distribution as X. The infinite sequence that I cited is by definition the same thing as X1.