My solution never mentioned Bayes! So, don't have to "be a Bayesian" to use what I wrote.

I just used conditional expectation which, of course, with full details, is from the Radon-Nikodym theorem in measure theory as in, say, Rudin, Real and Complex Analysis with a nice, novel proof from von Neumann.

Yes, of course. A “Bayes prediction function” has nothing to do with Bayesian. Bayes had a lot of things named after him ;)