What does $x \sim X$ mean in probability? I just want to confirm what  $x \sim X$ mean in probability. What does the small $x$ and big $X$ represent? And can we replace $\sim$ with $=$.
Disclosure: I had read many probability textbooks from front to back such as Leon-Garcia and Papoulis, but never encountered this notation until I looked at some notes online.  I think this is a notation used in stastistics, because $\sim$ is not very informative in my view.
 A: Typically you would say something like the following 
$$ X \sim \textrm{Uniform}(-1,1) \tag{1} $$
then $X$ is a random variable and it follows the uniform distribution with parameters $-1$ and $1$. Typically a uniform distribution is like this. 
$$ f_{X}(x) =\begin{align}\begin{cases} \frac{1}{b-a} &  \textrm{ for } a \leq x \leq b \\ 0   &  \textrm{ everywhere else }  \end{cases} \end{align} \tag{2}$$
when with parameters $-1$ and $1$ we have
$$ f_{X}(x) =\begin{align}\begin{cases} \frac{1}{2} &  \textrm{ for } -1 \leq x \leq 1 \\ 0   &  \textrm{ everywhere else }  \end{cases} \end{align} \tag{3}$$
In other words, it is saying that $x$ is a random variable and following a distribution given by $X$ in your case, with a distribution function given by it. In other words, it is like the $=$ sign but typically it can refer to a family of functions.
Edit:
If you read the paper in section $3$ on page $2$ it literally defines things. 

To learn the generator’s distribution $p_{g}$ over data x, we define a
  prior on input noise variables $p_{z}(z)$,

It uses notation like this in $4$

The generator G implicitly defines a probability distribution $p_{g}$ as
  the distribution of the samples G(z) obtained when $z ∼ p_{z}$

...did you read the paper?
Edits:
For the second part if you read the paper, it is defined on page $2$. 

where $p$ is the distribution to learn and $q_{\theta}$ is the distribution
  defined by the implicit generator. The expectation is minimized over a
  parametrized class of functions 

and you note that $\otimes$ is the tensor product. 
