It is true that a distribution T whose distributional laplacian is zero, $\Delta T = 0$, is actually $T = T_f$ for some smooth harmonic function $f$. What is S, sorry?

-1 While in some sense I would actually like to see a really good answer to this, I feel I must discourage this question on the grounds that (from FAQ): "MathOverflow is not the appropriate place to ask somebody to write an expository article for you" and that this is a little bit along the lines of (from how to ask): "what's the deal with algebraic geometry?". So to continue quoting from these advice pages: "The great answer you're hoping for doesn't exist because there isn't a precise question". But you could have a look at arxiv.org/abs/math/0304396

+1. Yes!; well-articulated. I do this all the time when talking with my girlfriend and she nearly always argues that the extremity of my example makes it invalid and irrelevant to the discussion, whereas I see the extremity as potentially highlighting the salient points of the general case.

You seem to be asking for suggestions without much motivation. The feeling of MO is that questions should have concrete answers unless specifically asked as big list communit wiki questions. FAQ reading is recommended, and then perhaps think of a way to ask a single question with a question mark at the end. You will already have made progress if you can do this.

@Deane If you're referring to the bit I think you're referring to then I am either still very confused or have failed to get my point across again: I have a connection on $P$ = principal connection, in the sense of smooth $G$- covariant choice of horizontal subspaces. This is equivalent to a one-form on $P$ with values in the lie algebra and some other properties. Then I discuss the possibility of another, different, connection in a vectore bundle $E \to P$. I was not supposed to suggest $\omega$ actually is said connection in $E$.

@Spiro, Thanks for your comment. I think my trouble comes from the fact that from the point of view of the manifold $P$, $\omega$ is just a certain one-form with values in some vector space which happens to be $\mathfrak{g}$. I see how this object is a connection in the bundle $P \to M$ but what does this have to do with differrentiating $\mathfrak{g}$-valued forms on $P$? [@Jose Corrected, thanks.]

Yes this seems to be precisely why finding such a measure will fail. If one wishes to weight all natural numbers equally, then only the trivial `zero measure' can assign a finite measure to an infinite set.