[cdavidcash]

>You don't know how much money is in either envelope, but it needn't have been selected randomly.

This is exactly wrong, at least in the mathematical formalization of this problem that everyone is assuming. The definition of "don't know how much money" must be taken to be "has a uniform distribution".

If you want to use some reasoning that treats the amount of money as an "unknown in an equation" then you are instead giving a solution that differs for each possible value of the money and does not capture the random choices we are trying to model.

By the way, no one said that uniform distributions on infinite sets don't exist. He said "infinite series (the list of possible monies)", a slightly mis-used term that clearly excludes [0,1], your chosen example.