Let $P$ and $Q$ be two probability measures over $R^n$, with CDF denoted by $F_P,F_Q$, respectively (that is, $F_P(x)=P(\{x'\in R^n:x'\leq x\})$, where $\leq$ is taken componentwise. The **Kolmogorov-Smirnov distance** between $P$ and $Q$ is defined as $d_{KS}(P,Q)=\sup_x |F_P(x)-F_Q(x)|$.

Let $f$ be a nonnegative, bounded Borel measurable function defined on **$R^n$**, say $0\leq f \leq M$, and denote by $E_P[f]$ the expectation (integral) of $f$ taken according to $P$; similarly for $E_Q[f]$. Suppose $d_{KS}(P,Q)\leq \epsilon$. What can be said about $|E_P[f]-E_Q[f]|$? Ideally, I am looking for a bound of the form $|E_P[f]-E_Q[f]|\leq \epsilon\cdot C$, for some constant $C$ not depending on $\epsilon$.