I was trying to solve question 4.3.1 from Katok and Hasselblat but I couldn't make my argument complete. Could you look at it?
This is the question.
Let $\xi, \eta \in \mathcal{P}_m$. Prove that for any $\epsilon > 0$ there is $\delta > 0$ such that $d_R(\xi, \eta) < \delta$ implies $D(\xi, \delta) < \epsilon$. Here
$d_R$ is the Rokhlin metric;
$$D(\xi, \eta) := \min_{\sigma \in \mathbb{S}_m} \sum_{i = 1}^n \mu(\xi \Delta \eta_{\sigma(i)});$$
$\mathcal{S}_m$ is the set of all permutations on $m$ elements; and
$\mathcal{P}_m$ is the set of all measurable partitions with exactly $m$ elements for the probability space $(X, \mu).
This is my thoughts/ideas (I overexplain things. Sorry in advance.)
