In this problem, the Hansdorff distance of two sets $A,B$ is $dist(A,B)=\inf\{\delta:A^{\delta}\subset B,B^{\delta}\subset A\}$, in which $A^{\delta}=\{x:dist(x,A)<\delta\}$
One of my ideas is to divide the compact sets in $\overline{B}$ into $C_I$ according to the size of the measure, where $\{C_I\}$ is countable and $C_{I}=\{A:A ~is~compact~in~\overline{B}~and~m(A)\in I\}$ , and prove that each $C_I$ is a sparse set (or direct proof prove that they are closed sets without interior points)
However, for the compact set sequence $A_n$ and compact set $B$ in $\overline{B}$ , if $dist(A_n,B)\to0$, then it is obvious that $m(A_n)\to m(B)$.
Therefore, for a compact set $B\in C_{(a,b)}$ , I cannot find a sequence of compact sets $A_n$ s.t. $A_n$ satisfies both $A_n\notin C_{[a,b]}$ and $dist(A_n,B)\to0$. So I cannot prove that $B$ is not the inner point of $C_{[a,b]}$.
And other approaches, such as dividing compact sets in $\overline{B}$ by their Hausdorff dimensions, are too complex to continue for me.