Integral representation and supports of functionals on Lipschitz spaces

Abstract

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.

v3: Upgraded several results in Sections 3.2, 5.2 and 6

References in corpus (1)

• The Samuel realcompactification of a metric space

Cited by in corpus (1)

• Extreme points in Lipschitz-free spaces over compact metric spaces