Adding a condition so the Lebesgue measure of closure is equal to original measure

Question

What assumptions could go in the blank so that this is a theorem? Are there any non-obvious ones?

Let $A\subset\mathbb{R}^n$ which is measurable and not necessarily closed, and assume __________________. Then $$\mathcal{L}^n\left(\overline{A}\right)=\mathcal{L}^n\left({A}\right).$$ (Here $\mathcal{L}^n$ denotes the Lebesgue measure on $\mathbb{R}^n$.)

Answer 1

I thought of an obvious one: Suppose the boundary of $A$ has measure zero. That’s unsatisfying, are there any others?