Continuity of Lebesgue measure w.r.t Hausdorff distance?

Question

If one considers the metric space $\mathcal{K}$ of all compact subsets of $\mathbb{R}^2$ endowed with the Hausdorff distance (that is $\Delta(A,B)=\inf \{ \delta: A\subset B^{\delta},B \subset A^{\delta} \}$). Is it true (and how can it be shown in that case) that the function $\mu: \mathcal{K} \to \mathbb{R}^2$ ($\mu$ is the 2-dimensional lebesgue measure) defined by $A \mapsto \mu{(A)}$ is continuous w.r.r the metric $\Delta$?