The measure of the limit of a sequence open sets

Question

Let $A_n \in [0,1]^k$ be a sequence of open sets in the Euclidean topology of $\mathbb{R}^k$ with the condition that $A_{n+1} \subset A_{n}$ for all positive integer $n$ and the limiting set $A = \lim_{n \rightarrow \infty} A_n$ has an empty interior, i.e. $A^\circ = \emptyset$.

I wonder if $A$ is a zero measure set?

Answer 1

No: let $a_n$ be an enumeration of the rationals in $[0,1]$, and take $A_n=(0,1)\backslash \{a_0, \ldots, a_n\}$.

Then $A$ is the set of irrationals in $[0,1]$ thus has empty interior yet full measure.