# sequences of plane measures converging to a singular one: terminology, etc

We are dealing with very "easy" sequences of uniform measures converging to singular measures (?), as in the following example: let $a$, $b$, and $c$ be vertices of a triangle in $\mathbb{R}^2$, and $a'$ be the point on the line $bc$ which is the orthogonal projection of $a$. Let $t\in (0,1]$ and define the $\mu_t$ to be the uniform probability measure supported on the triangle with vertices $ta+(1-t)a'$, $b$, and $c$.

Then one can define $\lim_{t\to 0}\mu_t$, which is (a kind of?) singular measure supported on an interval in the line $bc$. Or is it better to talk about generalized functions, as Dirac $\delta$-function is a 1-dimensional analog of this setting?

We have to manipulate such limits. What would be a good text to refer to for this kind of setup? (We certainly do not want to develop any theory like this from scratch...). Is there any standard terminology for, e.g., the direction orthogonal to the line $bc$? (It is tempting to call it "singular direction".)

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we have a "toy" inverse moment problem for signed measures which are linear combinations of uniform measures supported on triangles, which have vertices in a given finite set $S$. It appears that we need to distinguish "proper" triangles from these that are actually line segments (i.e. 3 vertices are collinear). The latter are singular w.r.t. to the Lebesgue measure, right? – Dima Pasechnik Oct 3 '12 at 17:08
What is known about the resulting distribution on the line $bc$ the resulting singular measure is supported on? We don't need this immediately, but might want (or need) to look into at some point --- unless this is known all along. – Dima Pasechnik Oct 5 '12 at 13:05