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Good point. Does this work over all infinite sets, though? Integers? Rationals?
It does work, actually if we're using Integers (there are as many integers as Rationals so we don't need to care about the latter set) we get the good old discrete probability distribution where we either have finite number of possibilities or at most countable infinity of possibilities, e.g set of all Integers.
Real numbers are strictly larger set than integers, so in continuous distribution we have in a sense more possibilities than countably infinite discrete distribution.
giving every possible arrangement of objects/atoms/information equal weight
Without an arbitrary upper bound on complexity, there are infinitely many possible arrangements.
Yvain said the finiteness well, but I think the "infinitely many possible arrangements" needs a little elaboration.
In any continuous probability distributions we have infinitely many (actually uncountably infinitely many) possibilities, and this makes the probability of any single outcome 0. Which is the reason why, in the case of continuous distributions, we talk about probability of the outcome being on a certain interval (a collection of infinitely many arrangements).
So instead of counting the individual arrangements we calculate integrals over some set of arrangements. Infinitely many arrangements is no hindrance to applying probability theory. Actually if we can assume continuous distribution it makes some things much easier.
In response to
Calibration for continuous quantities
Very useful considering that many variables can be approximated as a continous with a good precision.
Small nitpicking about "or any actual measurement of a continuous quantity". All actual measurements give rational numbers, therefore they are discrete.
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These initial weights are supposed to be assigned before taking into account anything you have observed. But even now (under the second interpretation in my list) you can't be sure that the world you're in is finite. So, suppose there is one possible world for each integer in the set of all integers, or one possible world for each set in the class of all sets. How could one assign equal weight to all possible worlds, and have the weights add up to 1?
I don't think that gets around the problem, because there is an infinite number of possible worlds where the energy state of nearly every subatomic particle encodes some valuable information.
By the same method we do calculus. Instead of sum of the possible worlds we integrate over the possible worlds (which is a infinite sum of infinitesimally small values). For explicit construction on how this is done any basic calculus book is enough.