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- W3217443154 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." @default.
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- W3217443154 date "2021-11-25" @default.
- W3217443154 modified "2023-09-23" @default.
- W3217443154 title "Integral Representation and Supports of Functionals on Lipschitz Spaces" @default.
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- W3217443154 doi "https://doi.org/10.1093/imrn/rnab329" @default.
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