This is just a quick technical note. Feel free to ignore. Bas van Fraassen defines a Popper probability measure as a Popper probability function that is defined on a σ-field and such that P(−|B) is a countably additive probability measure for any non-negligible B. We can say that B is negligible provided that P(∅|B)=1. One might hope to define a Popper probability function on, say, [0,1]2 such that if B⊆[0,1]2 is a closed non-self-intersecting smooth curve of non-zero length then B is in our σ-field, is non-negligible and P(−|B) is normalized Lebesgue length measure for Borel subsets of B.
I suspect that Popper probability measures aren't very interesting: there is too much that ends up negligible, thereby not making much of an improvement over ordinary probability measures.