all those ε’s…

A revealing [and interesting] question on X validated about ε’s… The question was about the apparent contradiction in writing Normal random variates as the sum of their mean and of a random noise ε in the context of the bivariate Normal variate (x,y), since using the marginal x conditional decomposition led to two different sets of ε’s. Which did not seem to agree. I replied about these ε’s having to live in different σ-algebras, but this reminded me of some paradoxes found in fiducial analysis through this incautious manipulation of ε’s…