Non-trivial subsets $A$ satisfying $m^{*}(A) = m^{*}(A \cap E) + m^{*}(A \cap E^c)$ for all $ E \subset \mathbb{R}$

Question

I was studying measure theory from Royden's book and here is the definition of a "Lebesgue measurable set" $E$:

Definition. A set E is said to be measurable provided for any set A, $$ m^{*}(A) = m^{*}(A \cap E) + m^{*}(A \cap E^c). \qquad (1)$$

Here the sets are subsets of $\mathbb{R}$ and $m^{*}$ is the Lebesgue outer measure.

I was wondering if there are sets with the roles of $E$ and $A$ reversed, i.e., are there sets $A$ such that for every set $E$, $A$ satisfies $(1)$? I can easily see that the sets $\phi$, $\mathbb{R}$ and countable sets satisfy this condition but I am unable to think of non-trivial examples.