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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