Epsilon-net of a set system

Remarks

Example: let $X = \set{1, 2, \dots, 100}$, let $\mathcal{R}$ be the sets of the form $\set{a, a+1, \dots, b}$, and let $\varepsilon = 1/10$. The set $N = \set{10, 20, \dots, 100}$ is an $\varepsilon$-net. Indeed, a range with at least $10$ elements consists of at least $10$ consecutive integers, so it contains a multiple of $10$.

Non-example: for the same $X$ and $\mathcal{R}$, the set $N' = \set{20, 40, \dots, 100}$ is not a $1/10$-net. The range $\set{1, \dots, 19}$ has $19 \ge 10$ elements but misses $N'$.

The same notion is used for a probability measure $\mu$ on a set $X$: then $N$ is an $\varepsilon$-net if it meets every range $R$ with $\mu(R) \ge \varepsilon$. The case above is $\mu$ equal to the uniform measure on a finite $X$.