I am looking for a succinct way to describe a subset of the natural numbers which has ``measure zero" in the following sense: Let X_1 \subset X_2 \subset .... be any strictly nesting sequence of finite subsets of natural numbers, I want to describe a subset S \subset \N which has the property that the limit of the usual counting measures applied to S restricted to X_i is zero.

I don't know that much about measure theory and am having trouble finding any measures on countable sets such as the natural numbers. There should be a natural measure on them such that the sets I describe above have zero measure with respect to it. Is there a name for such a natural measure on the natural numbers?

density, even if your definition is not the correct definition of density. In fact, you should divide the counting measure by the size of $X_i$ before taking the $\liminf$ (or the $\limsup$). See en.wikipedia.org/wiki/Natural_density for more details. $\endgroup$ – Francesco Polizzi Jun 12 '11 at 12:33