Monday, June 11, 2012

Absolutely nonmeasurable sets

The ideal of a non-zero (point) probability assignment to all possibilities is incoherent for cardinality reasons. Moreover, as Alan Hajek has insisted, the existence of nonmeasurable sets provides further difficulties.

One might try to get around both issues by problem-specific Bayesianism, where one only insists on a probability assignment specific to a particular problem at hand. This gets around my no-go theorem, since that theorem shows that there is no single non-zero probability assignment to all the possibilities there are. But in any given probabilistic calculation, the collection of possibilities is restricted to some set, and then there could well be a generalized probability (e.g., satisfying the axioms here) for that problem.

One might even have some hope that problem-specific Bayesianism could handle the issue of nonmeasurable sets. For there are isometrically invariant extensions of Lebesgue measure (i.e., extensions invariant under translation, rotation and reflection) that make some Lebesgue nonmeasurable sets be measurable.

But no such luck. Start by noting that there are absolutely nonmeasurable sets. A bounded absolutely nonmeasurable set (I'm making up this technical term) is a subset A of n-dimensional Euclidean space Rn such that there is no isometrically invariant probability measure that (a) makes A measurable, (b) assigns finite measure to every bounded measurable subset of Rn, (c) assigns non-zero measure to some bounded subset of Rn. The Hausdorff Paradox then shows that there is a bounded absolutely nonmeasurable set if n=3, assuming the Axiom of Choice.

In fact, from the Hausdorff Paradox we can prove that there is a bounded subset A of R3 such there is no isometrically invariant generalized finitely additive probability measure, e.g., in the sense of this post, on the cube [0,1]3 or on the three-dimensional ball of unit radius that makes A measurable.

So the problem-specific approach also runs into trouble, at least assuming the Axiom of Choice. And the Axiom of Choice (or, more weakly, the Boolean Prime Ideal Theorem--I don't know if this makes a difference, but in any case BPI has no intuitive support beyond the fact that AC implies it) is also assumed by hyperreal extensions of probability theory.

Of course, if one allows for interval-valued measures, that's different kettle of fish.


Alexander R Pruss said...

On absolutely nonmeasurable sets, see this paper.

Alexander R Pruss said...

Also, see MR0407247 (53 #11026).