This is a question on how to make sense of an equation (specifically, a description of a distribution) that, no matter how I slice it and dice it, always has some left over pieces that don't fit.
The equation in question describes a random variable $G$ drawn from a Dirichlet process $\textrm{DP}(\alpha_0,G_0)$: \begin{align} G = \sum_{k=1}^\infty \beta_k \delta_{\theta_k} \end{align} where $\alpha_0 > 0$ and $G_0$ is some base measure. The $\beta_k$ are stick breaking weights that depend on $\alpha_0$ and $\theta_k$ are atoms drawn from $G_0$.
The full description can be found at the top of page 4 of this paper.
My first question is whether I can read $G$ like a function. Say, if I have some "atom" (which term I don't fully understand) $\theta_j$, can I plug it in like: \begin{align} G(\theta_j) &= \sum_{k=1}^\infty \beta_k \delta_{\theta_k} (\theta_j) \newline &= \sum_{k \:\textrm{where} \: \theta_k = \theta_j} \beta_k \end{align} meaning if I happened to draw an atom that was identical to more than one $\theta_k$ I am just adding up the relevant weights $\beta_k$. And that the random variable $G$ itself can be loosely understood as very long vector of weights $(\ldots,G_j = \sum_{k \:\textrm{where} \: \theta_k = \theta_j} \beta_k,\ldots)$ such that it is just a repartitioning of the infinite dimension vector $(\ldots,\beta_k,\ldots)$.
But if I interpret it this way, I get into trouble because apparently, you're supposed to use the $G$ to draw the so-called "atoms" again. So I revise my understanding to think of $G$ as a multinomial distribution. But then the equation that sums over the infinitely many values of $\beta_k$ doesn't make any sense. A sum is a sum. How do you draw things from a sum, which has just a fixed scalar value?
Is my interpretation wrong? Whether it's wrong at the foundations or wrong in subtle ways, please correct me regardless.
I'm trying to make the leap from the prob/statistics in DeGroot's book to understanding measure theory and nonparametric processes and it's things like these that completely throw me for a loop.