I will follow up your references. Thanks for suggesting this interesting alternative.

I am not an expert in spectral theory, but my concerns are even more fundamental. Your answer starts "Yes,..." but I'm not sure what question you're answering, especially given my same confusion as Pinelis's. Could you state a theorem? I think that would make it much clearer to me what you are guaranteeing. Finally, given Pinelis's proof of the necessity and sufficiency of being in the interior, I am still confused by the claim of it being a "weaker" condition.

So how does the measure $\mu$ such that $\mu\{\frac 1 2 \} = 1$ fit into this?

So far, looks good. Need a bit more time to make sure I understand each step.

I don't see where your proof relies on the hypothesis that the moment sequence lies in the interior of $\mathcal M_k$?

Seems to be increasing. Do you mean the function g(x)=-f(x)?

Or "Hess, C. (1999). Conditional expectation and martingales of random sets. Pattern Recognition 32, 1543–1567."

Perhaps "Graf, S. (1980). A Radon-Nikodym theorem for capacities. J. Reine Angew. Math. 320, 192–214." might be a good launching off point.

You might look into the work by Van der Vaart.