Would you agree that there is no way to define probability consistently in an infinite space without using a cutoff? Not at all. Just take the Borel sigma-algebra of regions, and put any probability measure at all that you like on it. This gives you a consistent definition of probability.
Ok, I guess I should have used the word "unique" instead of "consistently". It's the arbitrariness here that's the issue. I should have said that there is no unique way to define probability on an infinite space.
Yes, you can consistently do whatever you want. The problem is that no matter how you do it, it's going to be arbitrary, right? So you might as well do it in a way that makes some physical sense, like looking at how statistics works in a physically local region of spacetime, and then generalizing from there. If you can think of a way that makes more sense, or is somehow less arbitrary, then I'm sure lots of people would be interested.
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timeasmymeasure
no subject
Date: 2010-09-30 01:34 am (UTC)Not at all. Just take the Borel sigma-algebra of regions, and put any probability measure at all that you like on it. This gives you a consistent definition of probability.
Ok, I guess I should have used the word "unique" instead of "consistently". It's the arbitrariness here that's the issue. I should have said that there is no unique way to define probability on an infinite space.
Yes, you can consistently do whatever you want. The problem is that no matter how you do it, it's going to be arbitrary, right? So you might as well do it in a way that makes some physical sense, like looking at how statistics works in a physically local region of spacetime, and then generalizing from there. If you can think of a way that makes more sense, or is somehow less arbitrary, then I'm sure lots of people would be interested.