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.
Tuesday, July 2, 2013
Popper probability measure
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.
Unfortunately, we can't do this. For consider the infinite-length non-self-intersecting curve B defined in polar coordinates by r=1−1/θ, for θ≥1. This would be non-negligible because finite subcurves of it would be non-negligible. Now chop up B into a sequence of curves of unit length each: B1,B2,.... At least one of these has non-zero probability given B, by countable additivity of P(−|B). But by the Lebesgue length measure condition and Popper function axioms, they must all have the same probability given B. And that will violate additivity of P(−|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.
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.
It's also not hard to prove that an isometrically invariant Popper measure on all Borel subsets of [0,1]^2 makes negligible every line segment.
ReplyDeleteSo, roughly speaking, we am here: Invariant Popper functions on [0,1]^3 make line segments negligible (see comment on an earlier post) while invariant Popper measures on [0,1]^2 make line segments negligible. Wonder how far one could improve on these results (are rectangles negligible for invariant Popper functions on [0,1]^3?).