My answer to your other question might also be relevant here: mathoverflow.net/a/470814/32507. It shows that $\varphi$ can be manipulated on a $\mu$-null set such that $\varphi^c$ becomes measurable.

If I remember correctly, something can be found in the book "Functional Analysis: Theory and Applications" by Robert E. Edwards. However, I do not have a copy at hand and I do not remember exactly which results are contained there.

@OscarLanzi: Do you claim that the cap is the set with the largest area? I guess that it is. Further, it would be interesting to look at all sets which are maximal under inclusion and then find the one with the minimal area. I guess that it is the "triangle" (positive orthant).

Which measure do you use on $\{u = 0\}$? Maybe one can do something, if one assumes $\nabla u \ne 0$ on $\{u = 0\}$ (in a certain sense). If $u$ has some further regularity, this will render $\{u = 0\}$ a submanifold.

One nitpick: for the improper function $g \equiv +\infty$ we have $g^* \equiv -\infty$ and $g^{**} = g \equiv +\infty$.

I am not totally sure, because I do not speak convex geometry fluently. But the set on the last line is nonempty and on the boundary of $A$. Hence, it is a face of $A$.