I am reading section 4 of this article about invariant measures:


Let $(X,d)$ a complete metric space, $\mathcal{M}$ the set of bounded Borel-regular measures on $X$. In section 4 they apply a fixed point theorem on a contraction map:$ (S,\rho):\mathcal{M}^1 \to \mathcal{M}^1$. They call the metric in this space of measures $\mathcal{M}$ the $L$-metric, being defined as follows: $$L(\mu,\nu) = \sup\{\mu(\phi)-\nu(\phi): \text{Lip} \ \phi \leq 1 \} $$

So $L(\mu,\nu) = \sup_{\phi\in U}\{\mu(\phi)-\nu(\phi)\}$ where $U$ is the set of Lipschitz-maps on $X$ with Lipschitz constant $L\leq 1$. In order to apply a fixed point Theorem on $\mathcal{M}^1$ (the set of $\mu$ with $\mu(X)=1$) we need to know that the space $(\mathcal{M}, L)$, or even $(\mathcal{M}^1,L)$ is complete.

Can someone provide some info where I can find more about why this is true. (maybe it is easy to show). Also I would like to know the name of this $L$-metric, cause I cant imagine it's its real name.

Also, whats the usefulness of knowing the existence of a measure $\mu$ with mass 1, such that $$\mu(\phi) = \sum_{i=1}^N\rho_i \mu(\phi\circ S_i) $$ ($\sum_{i=1}^n\rho_i = 1$, and $\{S_1,\cdots, S_N\}$ a set of contractions maps on $X$.)

for $\phi \in BC(X)$, which is the same as $$\mu(A)=\sum_{i=1}^N\rho_i\mu(S_i^{-1}(A)) $$

Thanks for any info on this.


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