A characterization of Borel measures which induce Lipschitz free space elements

We solve a problem of Aliaga and Pernecká about Lipschitz free spaces (denoted by $\mathcal F(\cdot )$):

Does every Borel measure $\mu $ on a complete metric space $M$ such that $\int d(m,0)\, d|\mu |(m) \lt \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 obtain a characterization of the Borel measures $\mu $ such that $\mathcal L\mu \in \mathcal F(M)$, which indeed implies inner-regularity for complete metric spaces. We also 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. Finally, we partially solve a problem of Aliaga on whether every sequentially normal functional on $\operatorname{Lip}_0(M)$ is normal.