From Wikipedia
Let $(M, d)$ be a metric space with its Borel sigma algebra $\mathcal{B} (M)$. Let $\mathcal{P} (M)$ denote the collection of all probability measures on the measurable space $(M, \mathcal{B} (M))$.
For a subset $A \subseteq M$, define the $ε$-neighborhood of $A$ by $$ A^{\varepsilon} := \{ p \in M ~|~ \exists q \in A, \ d(p, q) < \varepsilon \} = \bigcup_{p \in A} B_{\varepsilon} (p). $$ where $B_{\varepsilon} (p)$ is the open ball of radius $\varepsilon$ centered at $p$.
The Lévy–Prokhorov metric $\pi : \mathcal{P} (M)^{2} \to [0, + \infty)$ is defined by setting the distance between two probability measures $\mu$ and $\nu$ to be $$ \pi (\mu, \nu) := \inf \left\{ \varepsilon > 0 ~|~ \mu(A) \leq \nu (A^{\varepsilon}) + \varepsilon \ \text{and} \ \nu (A) \leq \mu (A^{\varepsilon}) + \varepsilon \ \text{for all} \ A \in \mathcal{B}(M) \right\}. $$
- I wonder what the purpose, motivation and intuition of the L-P metric are?
- Is the following alternative a reasonable metric or some generalized metric between measures $$ \sup_{A \in \mathcal{B}(M)} |\mu(A) - \nu(A)|? $$ If yes, is this one more simple and easy to understand and therefore maybe more useful than L-P metric?
A related metric between distribution functions is the Levy metric:
Let $F, G : \mathbb{R} \to [0, + \infty)$ be two cumulative distribution functions. Define the Lévy distance between them to be $$ L(F, G) := \inf \{ \varepsilon > 0 | F(x - \varepsilon) - \varepsilon \leq G(x) \leq F(x + \varepsilon) + \varepsilon \mathrm{\,for\,all\,} x \in \mathbb{R} \}. $$
I wonder how to picture this intuition part:
Intuitively, if between the graphs of $F$ and $G$ one inscribes squares with sides parallel to the coordinate axes (at points of discontinuity of a graph vertical segments are added), then the side-length of the largest such square is equal to $L(F, G)$.
Thanks and regards!