If we had a model:
$y=x\beta +\eta$
and assumed exogeneity, so $E[\eta|x]$=0, is the fact that x or treatment intensity is now uncorrelated with $\eta$ equivalent to saying that x is 'independent of potential outcomes?'
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Sign up to join this communityYes, if we call the model
$$Y = X\beta + \eta$$
"structural" or "causal", we can use it to define potential outcomes as
$$Y(x) = x\beta + \eta.$$
(I'm using upper-case letters $Y$ and $X$ for random variables, and lower-case $x$ for a realization or fixed constant).
Then assuming $E[\eta|X] = 0$, it follows that
$$E[Y(x)|X] = x\beta + E[\eta|X] = x\beta,$$
which does not depend on $X$, so mean ignorability $E[Y(x)|X] = E[Y(x)]$ holds.
This "structural definition of counterfactuals" was proposed by Judea Pearl, see for example his book "Causality", or his book with Jewell and Glymour, "Causality: A Primer".