What do you mean by a product topology here? The product topology being used for a stochastic processes? That requires a topology on the state space in the first place. Right now I have not specified any topologies.
Regarding the stochastic aspect, I have thought about that, but so far I have not yet seen a benefit by including it because any stochastic approach can somehow be seen as just a deterministic approach on the level of distributions. I.e. if a Model M is actually a random variable, and a task T is also a random variable, then the important thing,...
Let me try rephrase is in more conventional probability theory. You are looking at a metric space of universes . You probably want to take the Borel-sigma algebra as your collection of events. We think of propositions as sets , which really just means is a subset which is not too irregular. Then thebindicator function is if A holds in universe and otherwise.
Your elaborations do not depend much on the time so we set .
You now talk about picking a universe uniformly from a ball . This is a problem. On finite...
Although we can not rigorously say this yet since we have not chosen a definition of agent, I think this intuitively applies and therefore (H2) can only hold when you are restricted to some set of tasks, perhaps "reasonable tasks", yea.
I wonder if in the stochastic inteprretation of task this issue disappears because "No Free Lunch" tasks that "diagonalize against a model in a particular fashion have very low probability.