A Characterization of Borel Measures which Induce Lipschitz-Free Space Elements
Lucas Maciel Raad
TL;DR
This work characterizes when a Borel measure $\mu$ on a pointed complete metric space $M$ yields an element of the Lipschitz free space $\mathcal{F}(M)$ via $\mathcal{L}_\mu(f)=\int f\,d\mu$. The authors establish a sharp criterion: $\mathcal{L}_\mu\in\mathcal{F}(M)$ if and only if $\int \rho\,d|\mu|<\infty$ and $\mu$ is concentrated on a separable subset of $M$, which in the complete case implies inner-regularity. They further link a global property—whether every Borel measure induces an element of $\mathcal{F}(M)$—to the weight $w(M)$ being strictly below the least real-valued measurable cardinal, leading to an independence result from ZFC for the universal statement. The paper also connects these measure-theoretic findings to normality notions in $\mathcal{F}(M)^{**}$, showing that normality coincides with weak$^*$-continuity and highlighting potential ZFC limitations in constructing counterexamples related to sequential normality. Overall, the results sharpen the understanding of how measure-theoretic and set-theoretic cardinal phenomena govern the structure of Lipschitz free spaces.
Abstract
We will solve a problem by Aliaga and Pernecká about Lipschitz free spaces (denoted by $\mathcal F(M)$): $$\text{Does every Borel measure $μ$ on a complete metric space $M$ such that $\int d(m,0) d |μ|(m)< \infty$ induce a weak$^*$ continuous functional $\mathcal Lμ\in \mathcal F(M)$ by the mapping $\mathcal Lμ(f)=\int f d μ$ ? }$$ In particular, we will show a characterization of the measures such that $\mathcal Lμ\in \mathcal F(M)$, which indeed implies inner-regularity for complete metric spaces, and we will prove that every Borel measure on $M$ induces an element of $\mathcal F(M)$ if and only if the weight of $M$ is strictly less than the least real-valued measurable cardinal, and thus the existence of a metric space on which there is a measure $μ$ such that $\mathcal Lμ\in \mathcal F(M)^{**} \setminus \mathcal F(M)$ cannot be proven in ZFC.