The usual fomalization of probability through $\sigma$-algebras and $\sigma$-additive measures can effectively model (countable) infinite chains of trials. This is usually done by defining a countable sequence of measurable spaces $(E_n, \mathcal{F}_n)$ (that represent the $n$-th trial), a starting probability measure on $(E_0, \mathcal{F}_0)$ and a sequence of transition probability kernels $K_n$ from $(E_0 \times\dots\times E_{n-1}, \mathcal{F}_0\otimes\dots\otimes\mathcal{F}_{n-1})$ into $(E_n, \mathcal{F}_n)$ that formalize the conditional interactions between the subsequent trials.
Then the probability measure over $\bigotimes_{n \in \mathbb{N}} E_n$ that represents the overall experiment (i.e. such that its finite-dimension restrictions are coherent with the known kernels and the known starting probability measure ) exists and is unique by Ionescu-Tulcea's theorem.
Now my question is:
- Can we extend this formalization process (and Ionescu-Tulcea's theorem) to effectively model transfinite chains of trials?
That is, given a transfinite sequence of "measurable" spaces $(E_\alpha, \mathcal{F}_\alpha)_{\alpha \in \kappa}$, a starting "probability measure" on $E_0$, and a transfinite sequence of transition "probability" kernels (in the limit trials maybe we could work with direct limits of kernels...) can we find a space that represents the overall sequence of trials?
I used the quotes before as I don't think this can be done within the boundaries of standard probability theory. In fact we'd need to go beyond $\sigma$-algebras in order to be able to work with events of uncountable dimensions , i.e. that are assessable only after uncountably many trials (e.g. given an uncountable sequence of tosses of a highly unfair coin, we'd like to work with the events regarding the cofinality of heads or tails ).
Moreover we'd need perhaps to work with measures taking values in a field extension of the reals as we'd deal with events of infinitesimal probability, hence leaving the usual interpretation of probability off the table.
Has this been done in some way?
I ask this question because recently I have been thinking of how to model extremely rare random "events" that manisfest themselves consistently only in a (uncountable) transfinite sequence of trials.