The Kantorovitch Representation Theorem for the Dual of $L^{\infty}(X, \mu)$

Verify (26) and (27).

Show that $\mathcal{B} \mathcal{F} \mathcal{A}(X, \mathcal{M}, \mu )$ is a linear space on which $\left\| \cdot \right\|_{\text{var }}$ is a norm. Then show that this normed linear space is complete.

For $\Phi \in \left[L^{\infty }(X, \mu )\right]^{*}$, let $\mathcal{R}(\Phi ) \in \mathcal{B} \mathcal{F} \mathcal{A}(X, \mathcal{M}, \mu )$ be the representative of $\Phi$. Prove that $\mathcal{R}:\left[L^{\infty }(X, \mu )\right]^{*} \rightarrow \mathcal{B F} \mathcal{A}(X, \mathcal{M}, \mu )$ is an isometry.