I have been trying to make an expression in terms of a ``Dirac Delta function'' rigorous, but I failed miserably. Please help, if you can. References are welcome.
The motivation for this comes from calculating densities $\rho$ in a measure space $M$, if one is given `samples' $\alpha(\omega)$ indexed by $\omega$ in another measure space $\Omega$. This has applications in statistical physics, so perhaps people have done this already.
For starters, if $\left(\Omega, \mathcal A \right)$ is a measurable space and $\omega \in \Omega$, then the Dirac measure is $$ \delta_\omega \colon \, \mathcal A \mapsto \mathbb R \, , \quad A \to \delta_\omega\left(A \right) := \begin{cases} 1 \, , \, \omega \in A \\ 0 \, , \,\omega \notin A \end{cases} \, . $$ I believe that the problem at hand requires this concept, and not the Dirac distribution.
Now let $(M,\mathcal A, \mu)$, $(\Omega,\mathcal B, \nu)$ be $\sigma$-finite measure spaces and let $\alpha \colon \Omega \to \mathcal M$ be a measurable function. Then I want that for any $\theta \in M$ and any sequence of measurable sets $\left(N_n (\theta)\right)_{n \in \mathbb N}$ containing $\theta$, having finite measure and with $$\lim_{n \to \infty} \mu \left(N_n (\theta)\right) = 0$$ that $$\theta \mapsto \rho \left( \theta \right) = \lim_{n \to \infty} \frac{\nu \left( \alpha^{-1} \left(N_n (\theta)\right)\right)}{\mu \left(N_n (\theta)\right)}$$ is well-defined and measurable.
This is where the Dirac measure comes in: I'm quite sure that for another (arbitrary?) sequence $\left( A_n \right)_{n \in \mathbb N}$ of
measurable sets in $\Omega$ with finite measure and with
$$\lim_{n \to \infty} \nu \left( A_n^C \cap \Omega \right) = 0$$
(i.e. `$A_n$ converges to $\Omega$ in measure') that
$$\rho \left( \theta\right) \overset{!}{=} \lim_{n \to \infty}
\frac{\int_{A_n} d \nu \negthinspace \left( \omega\right) \, \, \delta\left( \theta - \alpha_{\omega}\right)}{\nu \left( A_n\right)}$$
The problem is 1) to turn this into a mathematically sensible expression in terms of the Dirac measure, and 2) to show equality with the above one. Note that neither $M$ nor $\Omega$ can be assumed to be finite measure spaces.
EDIT: Alright, let me clarify things a bit.
1) The above description is a formalization of the following problem: Say I decided to place an uncountable number of points in $M= \mathbb{R}^n$, but I decided to label them in a rather inconvenient manner. So for each label $\omega \in \Omega$ I have a point $\alpha_\omega \in M$. Now the problem is that, say, suddenly I don't care about the labels any more, but rather I want to know how these points are distributed in space with respect to some volume measure (e.g. the Lebesgue measure $\mu$). That is, I want a density. So I want to integrate over my label space (each label has equal weight, so I have a measure $\nu$ on it) and only count those points in which I'm evaluating the density at. The above expressions are what I came up with.
2) The last equation is purely symbolic. The point is to "count" only those points in $\Omega$ for which $\theta = \alpha_\omega$. So it's not really useful to consider it as the Dirac delta of a function. The sequence of sets is only there, because $\nu(\Omega)$ may be infinite.
3) Yes, the first "definition" is probably problematic. For instance, what if all $\alpha_\omega$ are equal? We want to get a possibly singular measure out... Almost everywhere existence is perfect. Radon-Nikodym only works for $\Omega = M$, which is too restrictive.