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".)