Return to Question

I have a question linked to Interplay of Hausdorff metric and Lebesgue measure. Let us denote as $\mathcal K(\mathbb R^n)$ the space of compact subsets of $\mathbb R^n$ endowed with the Hausdorff metric $\rho$ and let $\lambda$ be the $n$-dimensional Lebesgue measure on $\mathbb R^n$. I want to know if there are (sufficient) conditions under which the measure $\lambda$ is continuous w.r.t. $\rho$, that is $$ \lim_{k\rightarrow\infty}\rho(K, K_k)=0\qquad\Rightarrow\qquad \lim_{k\rightarrow\infty}\lambda(K_k)=\lambda(K).\qquad(\star) $$ I tried to search it in the books Fractal geometry by Kenneth Falconer and Functions of Bounded Variation and Free Discontinuity Problems by Ambrosio, Fusco and Pallara but I did not find anything. In the second book it is written that, in the case $n=2$, the Hausdorff measure (which is a rescaling of the usual $\lambda$ on $\mathbb R^n$) is lower-semicontinuous w.r.t. the Hausdorff metric along sequences satisfying a suitable uniform concentration property, but this is not what I am looking for.

Some helps? Do you have some references?

Thank You

I have a question linked to Interplay of Hausdorff metric and Lebesgue measure. Let us denote as $\mathcal K(\mathbb R^n)$ the space of compact subsets of $\mathbb R^n$ endowed with the Hausdorff metric $\rho$ and let $\lambda$ be the $n$-dimensional Lebesgue measure on $\mathbb R^n$. I want to know if there are (sufficient) conditions under which the measure $\lambda$ is continuous w.r.t. $\rho$, that is $$ \lim_{k\rightarrow\infty}\rho(K, K_k)=0\qquad\Rightarrow\qquad \lim_{k\rightarrow\infty}\lambda(K_k)=\lambda(K).\qquad(\star) $$ I tried to search it in the books Fractal geometry by Kenneth Falconer and Functions of Bounded Variation and Free Discontinuity Problems by Ambrosio, Fusco and Pallara but I did not find anything. In the second book it is written that, in the case $n=2$, the Hausdorff measure (which is a rescaling of the usual $\lambda$ on $\mathbb R^n$) is lower-semicontinuous w.r.t. the Hausdorff metric along sequences satisfying a suitable uniform concentration property, but this is not what I am looking for.

Some help? Do you have some references?

I have a question linked to Interplay of Hausdorff metric and Lebesgue measure. Let us denote as $\mathcal K(\mathbb R^n)$ the space of compact subsets of $\mathbb R^n$ endowed with the Hausdorff metric $\rho$ and let $\lambda$ be the $n$-dimensional Lebesgue measure on $\mathbb R^n$. I want to know if there are (sufficient) conditions under which the measure $\lambda$ is continuous w.r.t. $\rho$, that is $$ \lim_{k\rightarrow\infty}\rho(K, K_k)=0\qquad\Rightarrow\qquad \lim_{k\rightarrow\infty}\lambda(K_k)=\lambda(K). $$ I tried to search it in the books Fractal geometry by Kenneth Falconer and Functions of Bounded Variation and Free Discontinuity Problems by Ambrosio, Fusco and Pallara but I did not find anything. In the second book it is written that, in the case $n=2$, the Hausdorff measure (which is a rescaling of the usual $\lambda$ on $\mathbb R^n$) is lower-semicontinuous w.r.t. the Hausdorff metric along sequences satisfying a suitable uniform concentration property, but this is not what I am looking for.

Some helps? Do you have some references?

Thank You

I have a question linked to Interplay of Hausdorff metric and Lebesgue measure. Let us denote as $\mathcal K(\mathbb R^n)$ the space of compact subsets of $\mathbb R^n$ endowed with the Hausdorff metric $\rho$ and let $\lambda$ be the $n$-dimensional Lebesgue measure on $\mathbb R^n$. I want to know if there are (sufficient) conditions under which the measure $\lambda$ is continuous w.r.t. $\rho$, that is $$ \lim_{k\rightarrow\infty}\rho(K, K_k)=0\qquad\Rightarrow\qquad \lim_{k\rightarrow\infty}\lambda(K_k)=\lambda(K).\qquad(\star) $$ I tried to search it in the books Fractal geometry by Kenneth Falconer and Functions of Bounded Variation and Free Discontinuity Problems by Ambrosio, Fusco and Pallara but I did not find anything. In the second book it is written that, in the case $n=2$, the Hausdorff measure (which is a rescaling of the usual $\lambda$ on $\mathbb R^n$) is lower-semicontinuous w.r.t. the Hausdorff metric along sequences satisfying a suitable uniform concentration property, but this is not what I am looking for.

Some helps? Do you have some references?

Thank You