  We analyze the relationship between Borel measures and continuous linear
functionals on the space $\mathrm{Lip}_0(M)$ of Lipschitz functions on a
complete metric space $M$. In particular, we describe continuous functionals
arising from measures and vice versa. In the case of weak$^\ast$ continuous
functionals, i.e. members of the Lipschitz-free space $\mathcal{F}(M)$,
measures on $M$ are considered. For the general case, we show that the
appropriate setting is rather the uniform (or Samuel) compactification of $M$
and that it is consistent with the treatment of $\mathcal{F}(M)$. This setting
also allows us to give a definition of support for all elements of
$\mathrm{Lip}_0(M)^\ast$ with similar properties to those in $\mathcal{F}(M)$,
and we show that it coincides with the support of the representing measure when
such a measure exists. We deduce that the members of $\mathrm{Lip}_0(M)^\ast$
that can be expressed as the difference of two positive functionals admit a
Jordan-like decomposition into a positive and a negative part.
