Suppose we want to draw uniformly at random from the set of all positive real sequences with $\ell_p$ norm $ = 1$, for some given value of $p$. Does a uniform distribution on this set exist for all $p$, or for any $p$?
It would seem to be easy to do such a thing for $\ell_\infty$. We can draw each coefficient uniformly at random between $[0,1]$, and if we do this countably many times, with probability $1$ the supremum will be $1$. I am curious if something similar is possible for general $\ell_p$, in particular $\ell_2$ and $\ell_1$.
Lastly, this kind of thing is a very bizarre probability distribution. For $\ell_\infty$, we basically have a uniform distribution on a hypercube of countably infinite dimension. But what kind of "probability" are we even talking about here? Does this even count as a probability density function if we're talking about some kind of countably infinite multiple integral? Basically, does this kind of thing even exist?
FWIW, I ran into this while working on a statistics problem where we have a (in theory) countably infinite number of parameters, such that the entire vector must have $\ell_2$ norm equal to $1$, and we're trying to do something like a MAP estimation with a flat prior on all possible parameter vectors. None of this poses any kind of real life problem as we can just truncate the parameter vector at some large number of parameters instead, rather than having it be countably infinite, but it made me curious if my flat prior was a real prior distribution or just an improper prior. I'm curious if there's any basic theory for probability distributions on sequence spaces.