# Interpretation of statement over probability distributions and functions

Consider the following paragraph from this research paper:

To learn the generator’s distribution $$p_g$$ over data $$x$$, we define a prior on input noise variables $$p_z(z)$$, then represent a mapping to data space as $$G(z;θ_g)$$, where $$G$$ is a differentiable function represented by a multilayer perceptron with parameters $$θ_g$$. We also define a second multilayer perceptron $$D(x;θ_d)$$ that outputs a single scalar. $$D(x)$$ represents the probability that $$x$$ came from the data rather than $$p_g$$.

Keep the parameters aside for this discussion and consider only domain and range.

1. distribution $$p_g$$ over data $$x$$

I am thinking that the term distribution refers to probability density function $$p_g$$ over a tuple $$X = (X_1, X_2, X_3, \cdots, X_n)$$ of random variables. That means $$p_g$$ takes a particular assignment of random variables and gives a real value that signifies the density of probability at that point. So, $$p_g : X \rightarrow R$$

Assume that the domain $$X$$ is a set of all possible assignments of random variables.

1. prior on input noise variables $$p_z(z)$$

I am thinking that $$p_z$$ is also a probability density function from $$R^n \rightarrow R$$.

1. represent a mapping to data space as $$G(z;θ_g)$$

I am interpreting the dataspace as the set of all possible assignments of random variables i.e., $$X$$. So, $$G$$ takes $$R^n$$ as domain and $$X$$ as range.

1. $$D(x;θ_d)$$ that outputs a single scalar. $$D(x)$$ represents the probability

I am interpreting $$D$$ as a probability measure that takes $$R^n$$ as input (may be from dataset $$X$$ or random from $$R^n$$) and outputs an element from $$[0,1]$$.

Are my interpretations for 1, 2, 3 and 4 correct in case of continuous random variables?

Note: I am using $$X$$ as tuple $$(X_1, \cdots, X_n)$$ as well as Cartesian product of $$X_1,X_2, \cdots X_n$$ i.e., $$X = X_1 \times X_2 \times \cdots \times X_n$$ for notational convenience.