[EDIT: This all deals with measures, not semimeasures; see below.]
For P to dominate Q in the old sense means that its Bayes score can only be some constant worse than Q, no worse, regardless of environment. My definition here implies that, but I think it’s a stricter requirement.
Your definition is equivalent to the standard definition. Li and Vitányi say that P dominates Q iff there is some α∈R such that for any binary string x, we have αP(x)≥Q(x). Li and Vitányi’s “probability distributions” take a binary string as input while the probability distributions we are using ask for an element of a σ-algebra, but we can abuse notation by allowing a binary string to represent the event of that the random sequence starts with this string as a prefix.
Theorem. For any probability distributions P,Q on Cantor space and any α, the condition αP(x)≥Q(x) holds for all binary strings x iff it holds for all x∈Σ.
Proof. The direction ⟸ is immediate. For ⟹, suppose that αP(x)≥Q(x) for all binary string events x. We first show that this property holds for every event in the algebra A generated by the binary string events. We can write any x∈A as a disjoint union of binary string events x=⋃ni=1yi. Then, αP(x)=αn∑i=1P(yi)≥n∑i=1Q(yi)=Q(x), as desired.
We can extend this to the σ-algebra generated by A, which is just the Borel σ-algebra Σ on Cantor space, by the monotone class theorem. I’m not sure how much measure theory you know, but you can think of this as a form of induction on measurable sets; if a property holds on an algebra of sets A and it is preserved by countable monotone unions and intersections, then it holds on every set in the σ-algebra generated by A. For countable monotone unions, if we have x1⊆x2⊆…, then αP(∞⋃i=1xi)=supiαP(xi)≥supiQ(xi)=Q(⋃ixi). We can do the same thing for countable monotone intersections x1⊇x2⊇…; αP(∞⋂i=1xi)=infiαP(xi)≥infiQ(xi)=Q(⋂ixi).□
EDIT: I’m talking about probability distributions P rather than semimeasures. I don’t know how your definition of dominance is helpful for semimeasures though. The universal semimeasure M is defined on binary sequences, and I think that question of whether P dominates M depends on how you extend it to the whole σ-algebra. I expect that many reasonable extensions of M would satisfy your Theorem 1, but this post (in particular the proof of Theorem 1) doesn’t seem to choose a specific such extension.
[EDIT: This all deals with measures, not semimeasures; see below.]
Your definition is equivalent to the standard definition. Li and Vitányi say that P dominates Q iff there is some α∈R such that for any binary string x, we have αP(x)≥Q(x). Li and Vitányi’s “probability distributions” take a binary string as input while the probability distributions we are using ask for an element of a σ-algebra, but we can abuse notation by allowing a binary string to represent the event of that the random sequence starts with this string as a prefix.
Theorem. For any probability distributions P,Q on Cantor space and any α, the condition αP(x)≥Q(x) holds for all binary strings x iff it holds for all x∈Σ.
Proof. The direction ⟸ is immediate. For ⟹, suppose that αP(x)≥Q(x) for all binary string events x. We first show that this property holds for every event in the algebra A generated by the binary string events. We can write any x∈A as a disjoint union of binary string events x=⋃ni=1yi. Then, αP(x)=αn∑i=1P(yi)≥n∑i=1Q(yi)=Q(x), as desired.
We can extend this to the σ-algebra generated by A, which is just the Borel σ-algebra Σ on Cantor space, by the monotone class theorem. I’m not sure how much measure theory you know, but you can think of this as a form of induction on measurable sets; if a property holds on an algebra of sets A and it is preserved by countable monotone unions and intersections, then it holds on every set in the σ-algebra generated by A. For countable monotone unions, if we have x1⊆x2⊆…, then αP(∞⋃i=1xi)=supiαP(xi)≥supiQ(xi)=Q(⋃ixi). We can do the same thing for countable monotone intersections x1⊇x2⊇…; αP(∞⋂i=1xi)=infiαP(xi)≥infiQ(xi)=Q(⋂ixi). □
EDIT: I’m talking about probability distributions P rather than semimeasures. I don’t know how your definition of dominance is helpful for semimeasures though. The universal semimeasure M is defined on binary sequences, and I think that question of whether P dominates M depends on how you extend it to the whole σ-algebra. I expect that many reasonable extensions of M would satisfy your Theorem 1, but this post (in particular the proof of Theorem 1) doesn’t seem to choose a specific such extension.