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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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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 (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 or the Wasserstein metrics.

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  • $\begingroup$ 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? $\endgroup$
    – Dima Pasechnik
    Oct 3, 2012 at 17:08
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    $\begingroup$ Correct. BY the way: The former ones (uniform measures on "full" triangles) are absolutely continuous w.r.t. Lebesgue measure. Probably also Lebesgue's decomposition theorem (en.wikipedia.org/wiki/Lebesgue%27s_decomposition_theorem) could be helpful. $\endgroup$
    – Dirk
    Oct 3, 2012 at 18:20
  • $\begingroup$ 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. $\endgroup$
    – Dima Pasechnik
    Oct 5, 2012 at 13:05
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Billingsley's Convergence of Probability Measures seems to me to be the standard classical text on the topic.

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This is a very simple calculation. For a start, you can assume that $B=(0,0)$, $C=(1,0)$ and $A=(p,q)$. $A‘$ is thus $(p,tq)$. The computations are then very easy, or, if this is too difficult, one can reduce to the case $p=1$, i.e., that of a right triangle. I doubt if you will find a reference for anything this trivial—-you certainly don‘t require the monographs mentioned above.

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