Help explain this "atom" and what this $G$ is. Following are some slides from a lecture that you can watch here (starting at 20:40). It is explaining the Hierarchical Dirichlet Process. https://www.youtube.com/watch?v=PxgW3lOrj60
In the first slide, $\pi=(\pi_1,...,\pi_k)$ is the parameter of a multinomial distribution on a bunch of second-layer distributions characterized by their parameters $\phi=\phi_1,...,\phi_k$. Each time you choose a second-layer distribution first according to $pi$, and then you sample from the chosen distribution.So it is very obvious that the probability of a point $x$ being sampled is $p(x|\pi,\phi)=\sum\limits_k {{\pi _k}p(x|{\phi _k})}$. The following are slides explaining this simple idea.


Now here comes the problem. The lecturer claims there is another way to express the above idea (the third and the fourth slides) using "atom". The definition of "atom" from wikipedia is
Given a measurable space $(X, \Sigma)$ and a measure $\mu$ on that space, a set $A$ in $\Sigma$ is called an atom if $\mu(A)>0$ and for any measurable subset $B$ of $A$ with $\mu(B)<\mu(A)$ one has $\mu(B)=0$.
I understand this definition, but I still don't understand exactly what the later two slides are talking about. I hope someone can help. Thank you! 


 A: I think his use of "atom" is a little off. I agree with Wikipedia's definition: an atom is a set, not a measure. I think he means to say that the $\phi_k$ are the atoms.

$G$ is another way of expressing the distribution $\pi$. The intuition and concept is exactly the same; it is just obfuscated by terminology and notation.
$\pi$ is like a weighted $K$-sided die, with side $k$ having weight $\pi_k$. To think of this in terms of a measure $G$, you can define a probability space $X$ as $$X := \{\phi_1,\ldots \phi_K\},$$ a set of $K$ points (these will end up being the atoms of the probability space with respect to $G$), and define the measure $G$ on $X$ to give the $k$th point a measure of $\pi_k$. The $\delta_{\phi_k}$ is the Dirac delta; it is a function $\delta_{\phi_k}:X \to \mathbb{R}$ defined by $$\delta_{\phi_k} (\phi_j) := \begin{cases} 1 & {j=k}\\ 0 & {j \ne k}\end{cases}$$
So, $G$, also a function $X \to \mathbb{R}$, is simply defined by $G(\phi_j):= \pi_j$ for all $1 \le j \le K$. This is a measure on $X$ that is essentially the probability distribution $\pi$. If you "plot" $G$ with $X$ on the horizontal axis, you get the "picket fence" image that Jordan describes: each picket represents a point $\phi_k$ and the height of the picket is $\pi_k$.

Remark: I think his use of $\phi_k$ to mean two different things might be a potential source of confusion. At the beginning, he uses them to denote parameters, but later when defining $G$, he also uses them to represent the points of the probability space.
