Let $f$ be a piecewise affine concave function on $N_{\mathbb{R}}$ and $(\Pi(f),\Pi(f^{\vee}))$ the dual pair of polyhedral complexes associated to $f$. Denote by $\Lambda\mapsto\Lambda^{\ast}$ the correspondence $\mathcal{L}f$. Then

$${\mathcal{M}}_{\mu}(f)=\sum_{v\in\Pi(f)^{0}}\mu(\partial f(v))\delta_{v}=\sum_{v\in\Pi(f)^{0}}\mu(v^{\ast})\delta_{v}=\sum_{\Lambda\in\Pi(f^{\vee})^{n}}\mu(\Lambda)\delta_{\Lambda^{\ast}},$$

where $\delta_{v}$ is the Dirac measure supported on $v$.