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".) 



sequences of plane measures converging to a singular one: terminology, etc
Whether or not the limit is singular (e.g. with respect to the Lebesgue measure), there are several notions of convergence for measures which can reflect this. Probably the most simple one is [weak convergence of measures](http://en.wikipedia.org/wiki/Convergence_of_measures#Weak_convergence_of_measures) (I prefer the name "weak-\* convergence", because I find the view that measures form the "dual space of continuous functions" very helpful).


If you need to "quantify" the convergence in some way it could be helpful that the topology for weak convergence is metrizable is several cases and moreover, that there are different metrics which work, e.g. the [Prokhorov metric](http://en.wikipedia.org/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric) or the [Wasserstein metrics](http://en.wikipedia.org/wiki/Wasserstein_metric).


