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

We will solve a problem by Aliaga and Perneck\'a about Lipschitz free spaces (denoted by $\mathcal F(M)$): $$\text{Does every Borel measure $\mu$ on a complete metric space $M$ such that $\int d(m,0) d |\mu|(m)< \infty$ induce a weak$^*$ continuous functional $\mathcal L\mu \in \mathcal F(M)$ by the mapping $\mathcal L\mu(f)=\int f d \mu$ ? }$$ 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 $\mathcal L\mu \in \mathcal F(M)^{**} \setminus \mathcal F(M)$ cannot be proven in ZFC.