Integral Representation and Supports of Functionals on Lipschitz Spaces

Abstract

Abstract We analyze the relationship between Borel measures and continuous linear functionals on the space $\textrm{Lip}_0(M)$ 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, that is, 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 $\textrm{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 $\textrm{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.