Observables can clearly be extended to infinite partitions, and maybe further to a sigma algebra or something similar. One might want to also think of A and E as sigma algebras.
Observables can also be extended to talk about separating two subsets of W, rather than separating a subset of W from its complement. One could also talk about observables that don’t allow for arbitrary functions from the observed to A, but instead allow for some restricted class such as continuous or Kakutani functions.
Such restricted classes might make more sense when using this more general notion of observables, or it might be possible to entirely construct these classes from this notion of observables.
This could allow the theory to encompass game theory, since you could have two agents which choose a probabilistic strategy, while knowing the probabilistic strategy chosen by the other player.
Generalizing observability
Observables can clearly be extended to infinite partitions, and maybe further to a sigma algebra or something similar. One might want to also think of A and E as sigma algebras.
Observables can also be extended to talk about separating two subsets of W, rather than separating a subset of W from its complement. One could also talk about observables that don’t allow for arbitrary functions from the observed to A, but instead allow for some restricted class such as continuous or Kakutani functions.
Such restricted classes might make more sense when using this more general notion of observables, or it might be possible to entirely construct these classes from this notion of observables.
This could allow the theory to encompass game theory, since you could have two agents which choose a probabilistic strategy, while knowing the probabilistic strategy chosen by the other player.