Here is Prob. 20, Chap. 2, in the book Real Analysis by H. L. Royden and P. M. Fitzpatrick, 4th edition:
Let $E$ have finite outer measure. Show that $E$ is measurable if and only if for each open bounded interval $(a, b)$, $$ b-a = m^* \big( (a, b) \cap E \big) + m^* \big( (a, b) \setminus E \big). \tag{1} $$
By definition of measurability, $E$ is measurable if (and only if), for any set $A$, we have the equality $$ m^*(A) = m^*(A \cap E) + m^* \left( A \cap E^c \right), $$ where $E^c = \mathbb{R} \setminus E$, that is, $$ m^*(A) = m^*(A \cap E) + m^* ( A \setminus E). $$
And, we also have $$ m^* \big( (a, b) \big) = b-a. $$
In the light of the observations in the preceding two paragraphs, is it necessary here for set $E$ to have finite outer measure?
More specifically, can we not state the following?
The set $E$---whether of finite outer measure or not---is measurable if and only if for any bounded open interval $(a, b)$, the identity (1) holds.