ECorollary 6.9 in A Guide to NIP theories by Pierre Simon proves the following

Theorem.For every positive integer $k$ and every positive real $\varepsilon$ there is an integer $n=n(k,\epsilon)$ such that any probability measure $\mu$ on a finite set system $(X,\Delta)$ with VC-dimension $\le k$ admits an $\varepsilon$-approximation of size $\le n$.

Recall that a finite set system is a pair $(X,\Delta)$ where $X$ is a finite set and $\Delta\subseteq{\mathcal P}(X)$. Given a probability measure $\mu$, an $\varepsilon$-approximation is a set $B\subseteq X$ such that

$$\left | \frac{|S \cap B|}{|B|} - \mu(S) \right | \leq \epsilon.$$

holds for every $S\in\Delta$.

**Question** I can prove the theorem if working with multi-sets (i.e. sets whose elements are counted with some finite multiplicity). As it is stated it may not be true. Suppose $X$ contains two points of mass $p_0<p_1$ with $\varepsilon<p_0$ and $2\varepsilon<p_1-p_0$ it seems there isn't any $\varepsilon$-approximations, no matter what size.

Am I reading things wrongly?

EDIT1: I just realize that the proof of the main theorem of Section 6 (the (q,q)-theorem) can be adapted to work with multi-sets. So it is not a major problem. Still, multi-sets make the proofs messier and I would like to know if they are really necessary.

EDIT2: I just realize that in an earlier version of A Guide to NIP theories Corollary 6.9 was stated with $\epsilon$-net for $\epsilon$-approximation. This version is correct. If one is not interest in good bounds, one can obtain an $\epsilon$-net from a multi-set just forgetting multiplicity. In the proof of the (q,q)-theorem only $\epsilon$-nets are used.