Metrics for sets

Question

Other than the Hausdorff metric, are there any common/useful metrics for sets? I'm having a bit of trouble finding any, though maybe I'm searching for the wrong things.

I'd also be interested in examples of metrics on multisets!

Answer 1

There are many useful metrics. Here are some examples: Let $X$ be any nonempty set and let $^\omega X$ be the set of all functions $f \colon \omega \to X$ (if you don't know what $\omega$ means - take $\omega = \mathbb N$). For $f,g \in ^\omega X$ let $d(f,g) = 2^{-n}$, where $n$ is minimal such that $f(n) \neq g(n)$ and let $d(f,g) = 0$ iff $f = g$. It's easy to see that this is a metric and in fact, it is an ultrametric. The topology induced by $d$ is the product topology on $^\omega X$, if we start with the discrete metric on $X$.

Speaking of the discrete topology: Let $X$ be nonempty and let, for $x,y \in X$, $$ d(x,y) = \begin{cases} 1 & \text{, iff x = y} \\ 0 & \text{, otherwise} \end{cases} $$ Then $d$ induces the discrete topology on $X$.

Let $(X_n, d_n)$ be metric spaces for $n \in \omega$. For $(x_n)_{n < \omega}, (y_n)_{n < \omega} \in \prod_{n < \omega} X_n$ let $$ d((x_n)_{n < \omega}, (y_n)_{n < \omega}) = \sum_{n < \omega} \frac{d_n(x_n, y_n)}{2^n(1 + d_n(x_n,y_n))} $$ Then $d$ is a metric on $\prod_{n < \omega} X_n$ and $d$ induces the product topology.

Fix a prime $p$ and let $\mathcal Q_p$ be the field of $p$-adic numbers. Then the $p$-adic metric $d_p$ makes $\mathcal Q_p$ into a local field and $\mathcal Q_p$ can be characterized as the completion of $(\mathbb Q, d_p \restriction \mathbb Q)$.

Let $(V,E)$ be a connected, undirected graph. For $u,v \in V$ let $d(u,v)$ be the length of the shortest path between $u$ and $v$ in $(V,E)$. This defines a metric.

Let $\mathcal L$ be the set of all Lebesgue-integrable functions $f \colon \mathbb R \to \mathbb R$. For $f,g \in \mathcal L$ let $$ d(f,g) = \int \mid f-g \mid d\mu, $$ where $\mu$ is the Lebesgue-measure on $\mathbb R$. $d$ is a metric and in fact a norm on $\mathcal L$.

The list goes on and on... Unfortunately, I don't know of any nice metrics on multisets (and as far as I am considered, multisets are just sets of the form $X \times \omega$.)