**Disclaimer.** This is follow up to the question https://math.stackexchange.com/q/3486130/168758.

Let $X=(X,d)$ be a **Polish** metric space equiped with the Borel $\sigma$-algebra and let $\mu$ be a nonnegative measure. For a nonempty measurable subset $A$ of $X$ and $\varepsilon > 0$, define the $\varepsilon$-enlargement of $A$ by $A^\varepsilon := \{x \in X \mid \text{dist}(x,A) \le \varepsilon\}$, where $\text{dist}(x,A) := \inf_{a \in A} d(x,a)$ is the distance of the point $x$ from the subset $A$. Finally, define $A' := X\setminus A$ and $A^{-\varepsilon} := ((A')^\varepsilon)'$.

Question 1.Under what general conditions on $\mu$ do we have $\mu((A^\varepsilon)^{-\varepsilon}) \ge \mu(\overset{\circ}{A})$ for every measurable $A \subseteq X$ ?

Of course, if $((A^\varepsilon)')^\varepsilon \subseteq \overline{A'}$ (the closure of $A'$), then the inequality holds. For example, take $A = [0, 1]$ in $X=(\mathbb R,|\cdot-\cdot|)$. Then for any $\varepsilon > 0$, we have $(A^\varepsilon)^{-\varepsilon} = (0, 1) = \overset{\circ}{A}$, the interior of $A$.

Question 2.Give an example of a metric space $X$ and measurable $A \subseteq X$ such that $\overset{\circ}{A} \not\subseteq (A^\varepsilon)^{-\varepsilon}$.

# Notes

- We may restrict the questions to closed sets $A \subseteq X$ if that helps.
- Any kind of insight which would help make progress on the above questions is more than welcome.