In this recent paper, in the introduction, the authors say:
"Disproportionate divisions are easily seen to exist when all the demands are rational... However, this does notimmediately demonstrate existence for irrational demands, since taking limits with respect to the Hausdorff metric, for example, does not preserve measure".
The context is a partition of the unit interval $[0,1]$ into sub-intervals. I am trying to understand the emphasized sentence. Does it mean that there exists an infinite sequence of subsets of $[0,1]$, say $X_1,X_2,\ldots,$, and another subset $X\subseteq [0,1]$, and a measure $\mu$, such that:
- The Hausdorff distance between $X_n$ and $X$ approaches $0$ when $n\to\infty$, but -
- The measure $\mu(X_n)$ does not approach the measure $\mu(X)$ when $n\to\infty$?
If so, what is an example of such a sequence?
In $[0,1]$ let $X_n$ be the finite set $$ X_n = \left\{\frac{k}{n} : 0 \le k \le n\right\} $$ Then $X_n$ converges to $[0,1]$ in the Haudsdorff metric, even though each $X_n$ has measure $0$ and $[0,1]$ has measure $1$.