Consider the following expression
$$E_{x\sim p_{data}(x)}[ f(x) ] $$
I am understanding it as
1) if X is a collection of a discrete random variable $(X_1, X_2,......, X_n)$, and x is generated from X by a particular assignment of all random variables in the tuple
Then
$$E_{x\sim p_{data}(x)}[ f(x) ] = \sum\limits_{x} f(x) p_{data}(x) $$
2) if $X$ is a collection of a continuous random variable $(X_1, X_2,......, X_n)$, and $x$ is generated from $X$ by a particular assignment of all random variables in the tuple
Then
$$E_{x\sim p_{data}(x)}[ f(x) ] = \int\limits_{x_1}\int\limits_{x_2} \cdots \int\limits_{x_n} f(x) p_{data}(x) dx_n \cdots dx_1$$
Is my interpretation exact? If wrong, where am I going wrong?