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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 $$ \pi (\mu, \nu) := \inf \left\{ \varepsilon > 0 ~|~ \mu(A) \leq \nu (A^{\varepsilon}) + \varepsilon \ \text{and} \ \nu (A) \leq \mu (A^{\varepsilon}) + \varepsilon \ \text{for all} \ A \in \mathcal{B}(M) \right\}. $$

  1. I wonder what the purpose, motivation and intuition of the L-P metric are?
  2. 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?
  3. 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!

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    Atleast one motivation for the Prokhorov metric is the metrization of weak convergence of measures. This topology is very used and fruitful for applications. It is not however the only useful metric on $P(M)$ that metrizises weak convergence. Wasserstein metrics, that arise from optimal transportation of measures, also metrizise weak convergence if $M$ is compact, or Polish and $d$ bounded.2012-04-19
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    @ThomasE.: Thanks! By Weak convergence, do you mean [this link](http://en.wikipedia.org/wiki/Weak_convergence_of_measures#Weak_convergence_of_measures)?2012-04-19
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    Yeah, exactly. The first equivalent expression from the list is usually being used as a definition for the case of probability measures. For arbitrary measures the test functions need to have a compact support as well. I.e. $(\mu_{k})$ converges weakly to $\mu$ if $\int_{S}fd\mu_{k}\to \int_{S}fd\mu$ for all continuous, compactly supported $f:S\to \mathbb{R}$.2012-04-19
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    I have developed my answer somewhat here: http://math.stackexchange.com/a/358095/488902013-04-18
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    One can write $L(F,G)$ slightly differently as $$L(F,G) = \inf\{\epsilon > 0| F(x) \leq G(x+\epsilon) + \epsilon \text{ and } G(x) \leq F(x+\epsilon) + \epsilon, \forall x\in \mathbb{R}\}$$ so that more clearly L-P metric is a generalization of Lévy's metric2015-04-06
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    Just a sidenote: the metric you defined in #2 is the total variation distance.2016-01-18

3 Answers 3