Let $\mathcal{F}$ be a family of functions $D\subseteq \mathbb{R}^n\rightarrow \mathbb{R}$. Depending on the characteristics of these functions there are a number of metrics that we would naturally associate with $\mathcal{F}$, namely $C^p$ metrics, $L^p$ metrics and metrics that involve Lipschitz constants.
I'd like to hear about other metrics --possibly less well known than the ones mentioned-- that we associate naturally* with families of functions with certain properties.
*By naturally I mean that the definition of the metric has something to do with the functions themselves, so not just metrics that you may consider say on any set of a given cardinality.
There are a number of interesting metrics on (metric) measure spaces, the simplest of which might be the discrepancy metric. Let $(\Omega, \rho)$ be a metric space and let $\mu$ and $\nu$ be two probability measures defined on $\Omega$. Let ${\cal B}$ be the set of all closed balls in $\Omega$. Then the discrepancy metric is defined as
$$d_D(\mu, \nu) := \sup_{B \in {\cal B}} |\mu(B) -\nu(B)|$$
Since $\mu$ and $\nu$ are probability measures we see immediately that $0\leq d_D \leq 1$, and (more importantly) that it is scale-invariant -- multiplying $\rho$ by any positive constant does not affect $d_D$. It's commonly used to study random walks on groups.