Why we use expectations instead of unknown density formulas

The chapter assumes that we can draw samples from the real-data distribution $P_{X}$ and from the generator-induced distribution $P_\theta$, even if neither density can be evaluated. For any integrable function h,

This is an exact mathematical identity. A finite-sample average is an estimate, not generally an equality.

Given independent identically distributed $X_1,\ldots,X_n$ drawn from P, the empirical estimate is $\frac1n\sum_i h(X_i)$. Under the Law of Large Numbers assumptions (in particular, a finite expectation), it converges to $\mathbb{E}_P[h(X)]$ as n grows. Individual estimates fluctuate.