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@AnuragSahay Thank you for the pointer towards Baker--Harman--Pintz! I am interested in asymptotics, so I guess stopping at Heath-Brown is all I can do. Also thanks for your second comment, in principle I agree with you, but I think it is not very important if we write out the initials or not.
I understand it this way: The "cost of redistribution" in your second paragraph is that the planner is restricted in its dictation of consumption rights/working requirements, since if those are chosen wrongly, people in the population will "cheat the system" by faking their talent. This is a very interesting quantification of cost of redistribution! What I originally thought of, however, was an "ideological cost", in the sense that it is a violation meritocracy to redistribute wealth that was earned through a free market system.
Thank you for this highly interesting answer! It doesn't quite answer my question because, as you say, wealth distribution is assumed to be costless throughout your model (which is an assumption that I am trying to replace by something more realistic).
@StevenLandsburg I am not familiar with any of this literature, that's why I am asking this question. (I have added the tag reference-request to make this more clear.) Could you recommend me a paper which derives the cost of redistributing wealth from other assumptions?
If $f:\Omega\to X$ is measurable (in the preimage sense), and $g\in X^*$, then for any Borel-measurable $B\subset\mathbb R$, we have that $g^{-1}(B)$ is $\text{Borel}(X)$-measurable, so $(g\circ f)^{-1}(B)=f^{-1}(g^{-1}(B))$ is $\Omega$-measurable. So we have weak measurability. Am I missing something?
This can also be explained from the fact that $$\mathsf E(X\mid\mathcal F)(\omega)=\mathsf E(X\mid A) \text{ for almost all }\omega\in A$$ is, in general, false for an $L^1$-random variable $X$, a sigma-algebra $\mathcal F$ and $A\in \mathcal F$. It becomes more obvious when writing for instance $$P(X_4>0 \ |\ X_3\in\mathbb R, X_2>0) = P(X_4>0\ |\ X_3\in\mathbb R),$$ which is, of course, false in general.
