The Lebesgue measure could be a candidate for the measure you are asking for. Creating a bounding box around the convex set in question and then scaling, we get a probability measure (measure of the entire set is 1). Some sorts of random walks inside this bounding box will converge to this scaled Lebesgue measure. I think this is the main idea behind the convex volume estimation algorithms.
The oracle you mention is the indicator function of the convex set. Instead of integrating this indicator function directly over Lebesgue measure to get volume, we can use MCMC to get the "expectation" of the oracle.
It's not obvious if we can realize the Riemann zeta function as an integral over a finite measure space, but if we could, I think MCMC could be used to evaluate it.
Yes, what you just wrote is evident. I'm saying, in response to your assertion above, that it is not trivial to derive a useful result about approximating the volume of convex sets with MCMC, and not at all obvious that anything smarter than independent probes will be useful. I happened to read the original 1991 paper on this topic during my PhD, so I'm still vaguely familiar with it.
To clarify: It's clear than you can approximate a convex volume with exponentially-many independent (random or deterministic) probes. That's therefore not an interesting problem.
Surprisingly: there is NO deterministic algorithm that can solve this problem in polynomial time. But, surprise #2: a randomized algorithm using MCMC can. Pretty cool.
So, this is the relevance of MCMC to the convex volume problem. Used cleverly, it can take you from exponentially-many probes to polynomial.
And in particular: You talk about "creating a bounding box" for the reference Lebesgue measure -- saying that this first step is easy is basically assuming the whole problem is solved. Because: How do you create a bounding box around a convex set in R^d, with polynomially-many probes, so that the error is bounded? You can do it with exponentially-many probes (just go out along each orthogonal axis). But that's not interesting -- if you have that many probes, forget about MCMC, just use a deterministic algorithm.
The oracle you mention is the indicator function of the convex set. Instead of integrating this indicator function directly over Lebesgue measure to get volume, we can use MCMC to get the "expectation" of the oracle.
It's not obvious if we can realize the Riemann zeta function as an integral over a finite measure space, but if we could, I think MCMC could be used to evaluate it.