Let me amend that–on any set with a finite measure, you can create a uniform distribution rather easily. Just take the measure of your subset and divide it by the measure of the set.
However, my introduction to measure theory never covered anything larger than the reals. I assume that you can have measurable sets in a case like that, but it’s not my specialty.
I hate it when my threads go off into things I don’t understand, and expecially when they aren’t even hijacked to do it!
“It should be possible to explain the laws of physics to a barmaid.” ~ Albert Einstein ~
“You should see the place where Einstein used to drink!” ~ Triskadecamus ~
I don’t understand much that is being discussed in this thread, but apparently neither does the OP, so I’ll address the OP directly.
I think you’re right. Consider only the positive integers. There are an infinite number of them, but they have a starting point at zero. (Or is it one? Doesn’t matter.)
So imagine an infinite line. Now, where is the “center”? I’m pretty sure that the center of an infinite line separates two infintely long sections. So the center point of the pool of available random choices is infinitely far from zero.
Thus, my conclusion is that if you choose a bunch of truly random positive integers, they’d average out to be infinitely large.
But, of course, noen of this can possibly be correct, and I think my head just exploded. I should have taken your recommendation to heart.
I’m not sure it’s possible to have a ‘random function’ with infinite domain and/or range… I don’t believe much work has been done even on random functions with finite parameters. It would be interesting to work on that problem… (what expectations you have for generating a random function from the integers [1-10] to [1-10].)
And, on a practical level, I’m not sure that it’s possible to select a uniform random real number from [0,1], or even a uniform rational number from [0,1]. (Rationals being the subset of the real numbers that can be expressed as an integral fraction x divided by y where x and y are both integers… also can be expressed as finite or repeating-loop decimal fractions.)
Most of the time, ‘generate a random fractional number from 0 to 1’ is actually implemented with something along the lines of ‘generate a random integer from 0 to 100 billion, and then divide by 100 billion.’ Much as the OP indicated, if you were truly generating random rational numbers between 0 and 1, you would almost all the time get a result that requires at least 10 thousand digits to precisely specify. And if you were truly generating random REAL numbers between 0 and 1, IIRC you would almost always get an obscure transcendental that can never even be completely described, like pi or e, except without a real rule that can be used to derive them.
Not sure about that last part, I remember reading a proof that there are many more irrationals that rationals between 0 and 1, based on the idea that the real numbers are uncountable but the rationals are countable… not sure if there’s a consensus on ‘how many’ transcendentals there are versus the other irrationals, which I can’t remember the word for if there is any… irrationals that can be derived from rationals using root expressions, such as the fifth root of one quarter.
It’s not a question of being able to implement a function–it’s enough to know that one exists with the right properties.
btw, there are uncountably many transcendentals and only countably many algebraic numbers.
Perhaps surprisingly, it’s actually harder to produce a uniform distribution on the rationals from [0,1] than it is to produce one on the reals in [0,1]. In fact, it’s impossible to do for the rationals, since they’re countably infinite. There’s a one-to-one mapping between the rationals in [0,1] and the integers, so if one had a uniform distribution on those rationals, one could use that mapping to produce a uniform distribution on the integers. But that’s impossible.
If your brain isn’t fried yet, by the way, how about this: Consider the set of all numbers that can be specified in any way whatsoever. This includes specifications like
“the number x such that the integral from 0 to x of exp(3y[sup]2[/sup] +2y[sup]pi[/sup]) dy = 1”
, or
“The exact number of grams of material in the Milky Way Galaxy”.
The set of all specifiable numbers is countable, and therefore vastly smaller than the set of all real numbers. So you’re correct, with a uniform distribution on the reals in an interval, you’re overwhelmingly likely to get a number that can’t even be specified.
And certainly, there exist distributions on sets larger than the reals. For instance, for the set of functions from (0,1) to itself, I can define a distribution such that the probability of picking “f(x) = 1/2” is 1, and the probability of any other function is zero. But of course, that distribution is not remotely close to being uniform.
I was expecting a “smallest number that can’t be described in nine words” paradox here, but funnily enough there doesn’t seem to be one 
Do we have a definition of uniform for such a distribution? What things have to have the same probability? To parallel the “sets of the same width (aka standard measure) have the same probability” we’d need a standard measure on this space.
Or are we proposing that a probability distribution where every point has probability zero is enough?
No, that describes all the standard probability distributions on R[sup]n[/sup].