A Characterization of Borel Measures which Induce Lipschitz-Free Space Elements

We will solve a problem by Aliaga and Pernecká about Lipschitz free spaces (denoted by $\mathcal{F}(M)$): $$\text{Does every Borel measure }\mu \text{ on a complete metric space }M\text{ such that }\int d(m,0)d\mathrm{\mid }\mu \mathrm{\mid }(m)<\mathrm{\infty }\text{ induce a weak}{}^{\ast }\text{ continuous functional }\mathcal{L}\mu \in \mathcal{F}(M)\text{ by the mapping }\mathcal{L}\mu (f)=\int fd\mu \text{ ? }$$ In particular, we will show a characterization of the measures such that $\mathcal{L}\mu \in \mathcal{F}(M)$, 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 $\mu$ such that $\mathcal{L}\mu \in \mathcal{F}(M{)}^{\ast \ast }\setminus \mathcal{F}(M)$ cannot be proven in ZFC.