Suppose we have a bounded open set $S$ in $R^n$. Consider the collection of dyadic cubes $C_k$'s (https://en.wikipedia.org/wiki/Dyadic_cubes). I was wondering if there always exists a finite positive integer $K$ such that (1) $S \subset \cup_{k=1}^K C_k$ and (2) $\mu(\cup_{k=1}^K C_k) < \mu(S)+\epsilon$, where $\mu$ is the Lebesgue measure in $R^n$.

Thanks very much!

Suppose we have a bounded open set $S$ in $R^n$. Consider the collection of closed dyadic cubes $C_k$'s (https://en.wikipedia.org/wiki/Dyadic_cubes). I was wondering if there always exists a finite positive integer $K$ such that (1) $S \subset \cup_{k=1}^K C_k$ and (2) $\mu(\cup_{k=1}^K C_k) < \mu(S)+\epsilon$, where $\mu$ is the Lebesgue measure in $R^n$.

Thanks very much!