Query: Infinite priors

[hicks@cs.titech.ac.jp]

I have the following question concerning the existence of certain priors.

Can we say that a uniform prior over an infinite domain exists?  For example,
the uniform prior over all natural numbers.  I wonder since it cannot be
expressed in the form p(n) = (f(n)/\sum_n f(n)), where f(n) is a well defined
function over the natural numbers.  In general, if f(n) is not a summable
series, then can the probability function p(n), whose elements have the ratios
of the elements f(n), i.e., p(n)/p(m) = f(n)/f(m), be said to exist?

I ask because a true Bayesian approach to some problems may require the prior
to be defined.  If there is no prior, then we can't say we are taking a
Bayesian approach.  If an infinite uniform prior does not exist, then we
cannot take the approach that "no prior knowledge" = "infinite uniform prior".
I.e., it would imply that any Bayesian approach involving the prior MUST begin
with some assumptions about the prior (i.e., it must be formed from a
summable/integrable function).

References or opinions would be welcome.  Craig Hicks. hicks at cs.titech.ac.jp