Popper Functions, Uniform Distributions and Infinite Sequences of Heads

Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P(A|B) = P(A ⋂ B)/P(B). A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under plausible symmetry conditions, Popper functions often fail to do what they were supposed to do. For instance, suppose we want to define the Popper function for an isometrically invariant case in two dimensions and hence require the Popper function to be rotationally invariant and defined on pairs of sets from some algebra that contains at least all countable subsets. Then it turns out that the Popper function trivializes for all finite sets: P(A|B) = 1 for all A (including A = Ø) if B is finite. Likewise, Popper functions invariant under all sequence reflections can't be defined in a way that models a bidirectionally infinite sequence of independent coin tosses.