How to "iterate this process into the limit"? Does it not contradict the first answer to this question of mine?

@MiGang I think you are right but all we need is $A\subseteq B'$.

@RobertFurber Any reference for Lebesgue's view?

When does the projective code (if that is a thing) for a nonmeasurable set appear?

Isn't "$V^{\mathbb{B}}\models$ the Riemann hypothesis holds" such a statement? Or are you looking for some "functional analytic" statement about the space $C^+(St(\mathbb{B}))$?

Am I right that $\mathbb{R}$ can't be a nontrivial ultraproduct over any index set? If the ultrafilter $U$ is countably incomplete then the result is $\aleph_1$-saturated, hence not $\mathbb{R}$. If it's countably complete then it's $\kappa$-complete for some measurable cardinal $\kappa$. If measure one many fields are non-Archimedean then so is the ultraproduct. If measure one many are Archimedean then they are subfields of $\mathbb{R}$, so by $(2^\mathfrak{c})^+$-completeness the ultraproduct is trivial.

This seems to work, but you need to explain: why does a branch that extends $\sigma$ and has supremum $q+\sup\sigma$ exist for each $q$? I think that's the purpose of choosing those $\omega$-sequences. See also Theorem III.5.12 of Kunen or Theorem 9.16 of Jech. They (as well as Roitman, and your proposed construction) are all essentially the same, up to minor notational variations

Is the rescaling $\mu_\lambda$ necessary?