Let me try rephrase is in more conventional probability theory. You are looking at a metric space of universes (U,d). You probably want to take the Borel-sigma algebra B as your collection of events. We think of propositions as sets A∈B, which really just means A⊂U is a subset which is not too irregular.
Then thebindicator function χA(u) is 1 if A holds in universe u and 0 otherwise.
Your elaborations do not depend much on the time so we set t=0.
You now talk about picking a universe uniformly from a ball Bϵ(u0)=u∈U:d(u,u0)<ϵ.
This is a problem. On finite dimensional vector spaces we have the lebesgue measurenand we can have such a uniform distribution. On your metric space of universes it is entirely unclear what this means. You have to actually specify a distribution. This choice of distribution then influences your outcome to the extreme.
It is similar to how you can not uniformly pick a natural number.
So here your result will be strongly influenced by the distribution.
What we can do is say the following: We fix a sequence of probability measures ρn on U so that ρn converges to δu0 in the sense of weak convergence of probability measures. What this means is that you choose a sequence of distributions which approximate the dirac delta at u0, the distribution which samples to u0 with probability 1.
Then you can say something like: “The butterfly probability decay sequence around u0 with respect to ρn is given by Pu∼ρn(u∈A).
Here I am also not formalizing your sense of “convergence in the middle” because this is extremely unlikely to correapond to somwthing rigorous. You can view the above as a sequence in n and then study it’s decay as n goes to infinity, which corresponds to ϵ going to zero.
But everything here will depend on your choice of ρn. You can not necessarily choose uniformly from a small neighbourhood in any metric space. If the metric space is an infinite dimensional vector space uniformly, this is not possible.
There may be an alternative which means you don’t have to choose the ρn. You can fix a metric betwern probability measures which metrizes weak convergence, for example the Wasserstein distance W. athen you could perhaps look at:
supρ:W(ρ,δu0)<ϵPu∼ρ(u∈A).
Let me try rephrase is in more conventional probability theory. You are looking at a metric space of universes (U,d). You probably want to take the Borel-sigma algebra B as your collection of events. We think of propositions as sets A∈B, which really just means A⊂U is a subset which is not too irregular. Then thebindicator function χA(u) is 1 if A holds in universe u and 0 otherwise.
Your elaborations do not depend much on the time so we set t=0.
You now talk about picking a universe uniformly from a ball Bϵ(u0)=u∈U:d(u,u0)<ϵ. This is a problem. On finite dimensional vector spaces we have the lebesgue measurenand we can have such a uniform distribution. On your metric space of universes it is entirely unclear what this means. You have to actually specify a distribution. This choice of distribution then influences your outcome to the extreme. It is similar to how you can not uniformly pick a natural number. So here your result will be strongly influenced by the distribution. What we can do is say the following: We fix a sequence of probability measures ρn on U so that ρn converges to δu0 in the sense of weak convergence of probability measures. What this means is that you choose a sequence of distributions which approximate the dirac delta at u0, the distribution which samples to u0 with probability 1. Then you can say something like: “The butterfly probability decay sequence around u0 with respect to ρn is given by Pu∼ρn(u∈A).
Here I am also not formalizing your sense of “convergence in the middle” because this is extremely unlikely to correapond to somwthing rigorous. You can view the above as a sequence in n and then study it’s decay as n goes to infinity, which corresponds to ϵ going to zero.
But everything here will depend on your choice of ρn. You can not necessarily choose uniformly from a small neighbourhood in any metric space. If the metric space is an infinite dimensional vector space uniformly, this is not possible.
There may be an alternative which means you don’t have to choose the ρn. You can fix a metric betwern probability measures which metrizes weak convergence, for example the Wasserstein distance W. athen you could perhaps look at: supρ:W(ρ,δu0)<ϵPu∼ρ(u∈A).
This may be infinite or zero though.