I wonder if the following is true:
- For any probability measure m on the real numbers, there is a possible world in which there is a process that generates an output quantifiable as a real number and where the chances of outputs precisely correspond to m.
If the answer is positive, then by
this, it is possible to have two different physical processes each of which "uniformly", i.e., in a translation-invariant way, "chooses" a number in [0,1], but the chances arising from the two processes will be probabilistically inequivalent. Indeed there will be a pair of subsets
A and
B of [0,1] such that there is no way of defining a chance that the first process lands in
A but there will be a definite chance that the result that the second process does, and vice versa for
B.
In the light of all the results like this on extensions of Lebesgue measure, one wonders:
- Which extension of Lebesgue measure (or measures derived from Lebesgue measure in standard ways) is the physics of our world governed by?
Or are chances for Lebesgue non-measurable outcomes all undefined?
Thinking about this extension of Lebesgue measure stuff made me wish that instead of proving stuff about Lebesgue measurable functions, mathematicians working in probability and analysis would keep track of which features of the Lebesgue measure they're using. Some proofs only need a sigma-finite complete translation-invariant measure on a sigma-algebra including the Borel sets, for instance. So they should be stated for such. Some may not need completeness. Other proofs actually need full Lebesgue measure. And some may need some other assortments of features, such as a sigma-finite complete translation-invariant measure on a sigma-algebra in which the Borel sets are dense (this lets you approximate sets by unions rectangles).
ReplyDeleteOf course, keeping track of so much would be a mess.