By Goodhart’s law, this set has the property that any will with probability 1 be uncorrelated with outside the observed domain.

If we have a collection of variables , and , then is positively correlated in practice with most expressed simply in terms of the variables.

I’ve seen Goodhart’s law as an observation or a fact of human society—you seem to have a mathematical version of it in mind. Is there a reference for that.

Almost equally hard to define. You just need to define U, which, by assumption, is easy.