Misunderstanding of dirac measure in conjunction with point process While trying to understand random measures and random counting measures, I find myself continually running into a misunderstanding of the dirac function. Although helpful, the answers for Question for understanding definition of point process lacked some explanation I find I need.
I understand that the dirac measure is defined as
$$\delta_{x}(A) = \left\{\begin{aligned}
&1 &&: x\in A\\
&0 &&: x\notin A
\end{aligned}
\right.$$
where $x\in X$, $A\in \Sigma$ and $(X, \Sigma)$ is a measurable space.
From Wikipedia, a point process or random counting measure is a random measure of the form
$$\mu = \sum_{n=1}^{N}\delta_{X_{n}}$$
where $\delta$ is the Dirac measure and $X_{n}$ are random variables.
My main issue is with $\delta_{X_{n}}$. Since $\mu$ is a measurable function from some probability space $(\Omega, F, P)$ to a set of measures on the borel $\sigma$-algebra of the complete separable metric space $X$, what is the interpretation of $\delta_{X_{n}}(\omega)$ where $\omega\in\Omega$?
 A: Upon further reading and discussion with another user in the comments section, I believe I have come up with a (somewhat) good understanding of the meaning of the dirac measure and the function
$$\mu = \sum^{N}_{n=1}\delta_{X_{n}}$$
The following definition is taken from https://en.wikipedia.org/wiki/Random_measure restricted to the case where $X=\mathbb{R}^{d}$, although it can be more general than this.
Given $X = \mathbb{R}^{d}$ for some $d\in \mathbb{N}$, $\mathcal{B}(X)$ the borel $\sigma$-algebra on $X$, and a probability space $(\Omega, \mathcal{F}, P)$, a random measure is a measurable function $\mu: \Omega \rightarrow \mathfrak{N}$ from $(\Omega, \mathcal{F}, P)$ to the measurable space $(M_{X}, \mathcal{B}(M_{x}))$, where $M_{X}$ is the set of all boundedly finite measures on $\mathcal{B}(X)$.
A random counting measure is a random measure $\mu$ of the form
$$\mu = \sum^{N}_{n=1}\delta_{X_{n}}$$
where $N$ is an integer-valued random variable (from $(\Omega, \mathcal{F}) \rightarrow (\mathbb{R}, \mathcal{P}(\mathbb{R}))$) and the $X_{n}$ are random variables (from $(\Omega, \mathcal{F}) \rightarrow (\mathbb{R}, \mathcal{B}(\mathbb{R}))$). For any $\omega\in \Omega$, we have
$$\mu(\omega) = \sum^{N(\omega)}_{n=1}\delta_{X_{n}(\omega)}$$
where $N(\omega)\in\mathbb{N}$ and for all $B\in \mathcal{B}(X)$
$$\delta_{X_{n}(\omega)}(B) = \left\{\begin{aligned}
&1 &&: X_{n}(\omega)\in B\\
&0 &&: X_{n}(\omega)\notin B
\end{aligned}
\right.$$
