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Stephen Tashi

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There can be no translation invariant probability measure on the non-negative integers. Yet it is possible to imagine a general class of sampling processes that "ought" to have a particular implementation that picks a non-negative integer at random and favors no particular integer.

( I'm aware that number theorists who want to talk about "picking a integer at random" have defined various pseudo-measures or "asymptotic measures" such as http://en.wikipedia.org/wiki/Natural_density. In case anyone was about to post that link, my question isn't about that topic. I'm not asking about how to pick an integer at random. I'm asking about a paradox. )

On the one hand, there can be no "uniform" probability distribution on the non-negative integers in the sense of a probability distribution satisfying 1) A singleton integer is a measureable set and 2) The measure of any measureable set is the same as that of any other set formed by adding a constant to each of the first set's members.

The impossibility follows from the countable additivity of measures, which implies the measure of the whole space would be the sum of the measures of each integer. We can't find any constant probability p that makes the sum add up to 1.

(This does not rule-out the possibility that there could be a translation invariant measure on the non-negative integers where a singleton integer is not a measureable set.)

On the other hand, consider the following general type of sampling process on the non-negative integers (which I mentioned in the thread https://www.physicsforums.com/showthread.php?t=637049 ).

Let M be the set of 1-to-1 mappings of the non-negative integers onto themselves. Let F be a 1-to-1 mapping between the elements of M and the real numbers in the interval [0,1]. Let G be a probability distribution defined on the non-negative integers (such as a Poisson distribution). Consider the following process of sampling from the integers. Pick an integer k from the distribution G. Pick a real number x from the uniform distribution on [0,1]. Find the map f = F(x). Let the value chosen from the integers be f(k).

The existence of the function F is established by the fact that the cardinality of the real numbers in [0,1] is the same as the cardinality of the set M.

Since there exists one such function F, we could create arbitrarily many other functions that accomplish a 1-1 mapping between M and [0,1] by exchanging the y-values among the (x,y) ordered pairs in F.

However, the non-existence of any translation invariant measure on the non-negative integers says that one or more of the following counter-intuitive ideas must be true.

1. Perhaps any F that exists would create a process that favored some integer more than another. This is counterintuitive because 1-1 mappings that are used in cardinality arguments are not restricted by an concerns about measure or metrics. They can be quite arbitrary. Why should there be some fundamental asymmetry in what we can accomplish with them? - especially since we are picking f= F(x) by using a uniform distribution for x on [0,1].

2. Perhaps any F that resulted in a translation invariant probability distribution would imply a probability measure where a singleton integer is not a measureable set. This is counterintuitive since the process is described in terms of selecting a single integer as a sample.

3. Perhaps there is a fundamental logical paradox in assuming that exact samples can be drawn from a uniform distribution on the real numbers. In practical terms it is impossible to select such a sample, but is there any contradiction involved in assuming it can be done theoretically?

4. Perhaps the assumption that "there exists an F" is not equivalent to saying that an F can ever be explicitly defined. To do a particular sampling process, we must define F explicitly. Can there be things that theoretically exist but that are also theoretically impossible to specify explicitly?