# What is a random sample (once again...)?

This is apparently asked many times but there is still something I do not understand. Taken from Mathematical description of a random sample:

Mathematical description of a random sample: which one is it and why?

1. $$X_1(\omega), X_2(\omega), ..., X_n(\omega)$$, where $$X_1, ..., X_n$$ are different but i.i.d. random variables.
2. $$X(\omega_1), X(\omega_2), ..., X(\omega_n)$$, where $$X$$ is a (single) random variable.

1. $$X_1(\omega_1), X_2(\omega_2), ..., X_n(\omega_n)$$, where $$X_1, ..., X_n$$ are different but i.i.d. random variables.
After a bit of research told me that nothing of the above is true; see for example the book of Dudley "Real analysis and Probability" (section 11.4). The empirical measure on a measure space $$(\Omega,\mathcal F,\mu)$$ is defined via a sequence of iid random variables $$X_i$$ defined on $$\Omega^{\mathbb N}$$, by the mapping $$A\mapsto \frac{1}{n}\sum_{i=1}^n\delta_{X_i(\omega)}(A)\text{ for }\omega\in \Omega^{\mathbb N}.$$ The approximation property says that for almost all $$\omega\in \Omega^{\mathbb N}$$, the empirical measure converges weakly (or weakly-$$*$$ depending on your moods ;-) ) to $$\mu$$.