3. Okay, so if I find some $C_M[T]$ such that $T_n$ is given as you state, I can use your observation to show the pushforward is not preserved. The passage states "non-linearity" culprit. In what way is the pushforward non-linear? (again for context, see above edit)

My follow ups are too long, so I edited the orginal post -- for context see question edits. 1. Is this statement/proof correct? 2. So compact support of $\mu_2$ wasn't essential? That statement in the highlighted passage made me first investigate using Uniform Boundedness Principle - was this a misleading statement then? Or is there another of showing a minimizer using this statement?

aha! ya got it, add it as answer so I can accept! thx

ya, plugging it in works. maybe i should have asked -- how can I see that's the solution. I tried using a direct method of calculus of variations and was getting mixed up.

This is good, I didn't follow everything -- but its the order of ideas. I can hunt for the details now.