Say there is a population of size $N$, e.g. the population of human beings in China. The distribution of a given characteristic of these individuals, e.g. height, follows a distribution $X$, with mean $\mu$. (Let us abstract for time dimension)
Now, I obtain an iid, random sample from $X$, of size $n$. The LLN tells me that:
$$ \lim_{n \rightarrow \infty} \frac{\sum x_i}{n} = \mu $$
But, we do not really need an "infinite" sample size to kill all the uncertainty in our sample estimate of the mean of $X$. Instead, it is sufficient to obtain a sample of size $N$, this is, to produce a census, because by definition:
$$ \mu \equiv \frac{\sum x_i}{N} $$
As it has been highlighted elsewhere, with population data, there is no inference. Thus, estimators are no probabilistic (abstracting from measurement errors, et cetera). Hence, shouldn't the LLN hold for finite sample size, rather than in the limit?
