Hutchinson metric

Consider only nonempty, compact, and finite metric spaces. For such a space ${\displaystyle X}$, let ${\displaystyle P(X)}$ denote the space of Borel probability measures on ${\displaystyle X}$, with

${\displaystyle \delta :X\rightarrow P(X)}$

the embedding associating to ${\displaystyle x\in X}$ the point measure ${\displaystyle \delta _{x}}$. The support ${\displaystyle |\mu |}$ of a measure in ${\displaystyle P(X)}$ is the smallest closed subset of measure 1.

If ${\displaystyle f:X_{1}\rightarrow X_{2}}$ is Borel measurable then the induced map

${\displaystyle f_{*}:P(X_{1})\rightarrow P(X_{2})}$

associates to ${\displaystyle \mu }$ the measure ${\displaystyle f_{*}(\mu )}$ defined by

${\displaystyle f_{*}(\mu )(B)=\mu (f^{-1}(B))}$

for all ${\displaystyle B}$ Borel in ${\displaystyle X_{2}}$.

Then the Hutchinson metric is given by

${\displaystyle d(\mu _{1},\mu _{2})=\sup \left\lbrace \int u(x)\,\mu _{1}(dx)-\int u(x)\,\mu _{2}(dx)\right\rbrace }$

where the ${\displaystyle \sup }$ is taken over all real-valued functions ${\displaystyle u}$ with Lipschitz constant ${\displaystyle \leq \!1.}$

Then ${\displaystyle \delta }$ is an isometric embedding of ${\displaystyle X}$ into ${\displaystyle P(X)}$, and if ${\displaystyle f:X_{1}\rightarrow X_{2}}$ is Lipschitz then ${\displaystyle f_{*}:P(X_{1})\rightarrow P(X_{2})}$ is Lipschitz with the same Lipschitz constant.[3]

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