Kantorovich–Rubinstein duality for endpoint masses on a compact interval

By [F1], $X=[0,1]$ is a compact metric space, so [F2] applies. The point $(0,1)$ is Borel by [F5], and [F3] makes ${\delta }_{(0,1)}$ a Borel probability measure. Its coordinate marginals are ${\delta }_0$ and ${\delta }_1$, so it is admissible; [F3] gives its cost as $d(0,1)=1$. Thus $W_1^X(\mu ,\nu )\le 1$.

Let $h$ be any feasible real-valued function, so it is 1-Lipschitz and hence continuous and Borel measurable. Its parts $h^{+}=\max \{h,0\}$ and $h^{-}=\max \{-h,0\}$ are nonnegative measurable functions by [F4]. Applying [F3] to each part, for $c\in \{0,1\}$ we get $\int h^{+}d{\delta }_c=h^{+}(c)<\mathrm{\infty }$ and $\int h^{-}d{\delta }_c=h^{-}(c)<\mathrm{\infty }$. Thus [F4] makes $h$ integrable against both endpoint masses and gives $\int hd{\delta }_c=h^{+}(c)-h^{-}(c)=h(c)$. In particular, $f(x)=-x$ is feasible and has integrals $f(0)=0$ and $f(1)=-1$, so its objective is $1$. Every feasible $g$ has objective $g(0)-g(1)\le \mathrm{\mid }g(0)-g(1)\mathrm{\mid }\le 1$, so the dual supremum is exactly $1$.

For any admissible coupling $\gamma$, the marginal conditions give $\gamma (\{0\}\times X)=1$ and $\gamma (X\times \{1\})=1$. The complement of their intersection has measure zero: each of the two complements is null, and finite subadditivity follows from the measure axioms [F6]. Their intersection is $\{(0,1)\}$. For any Borel $E$, if $(0,1)\in E$ then $\gamma (E)=1$, while if $(0,1)\mathrm{\notin }E$ then $E$ lies in the null complement of that singleton, so $\gamma (E)=0$; hence $\gamma ={\delta }_{(0,1)}$. Therefore every admissible cost is $1$, and step 1.1 gives $W_1^X(\mu ,\nu )=1$.

Steps 1.2 and 2.1 compute the primal and dual values as $1$, proving the example's claimed equality.