The Swiss Public-Use Sample of the national census is a 5% sample drawn from the entire census survey. According to the documentation, the persons are sampled uniformly without replacement. Persons are the "primary sampling unit", the selection probability and person weights are all equal: $\pi = 0.05, w = 1/\pi = 20$.
For each person there are household-level attributes as well. The sample is not uniformly distributed anymore when considering the households: Larger households have a bigger sampling probability than smaller households. In order to carry ot analyses at the household level (with variables such as "number of children in the household"), the records must be reweighted.
Intuition suggests that each person record $p$ should be weighted with $w_p / k_p$ (with $k_p$ being the number of persons in the household). This should hold exactly if the persons were drawn with replacement, and at least approximately in the "without replacement" scheme. (The sample size is by five orders of magnitude larger than the maximum household size.) However, I find it difficult to provide a formal argument for this.
Is my intuition correct? How would a proof or a probabilistic argument look like? Can you suggest a textbook that treats similar examples, perhaps even this?