Is $\lambda ^{*} (A \setminus B) \leq \lambda^{*}(A)-\lambda^{*}(B)$ with $B\subseteq A$ generally true?

Question

Is $\lambda ^{*} (A \setminus B) \leq \lambda^{*}(A)-\lambda^{*}(B)$ generally true? Where $\lambda^*$ is the Lebesgue outer measure, and I am assuming $B \subseteq A$.

It seems correct intuitively. Under which conditions does this happen? For all sets? Only Lebesgue measurable sets?

Answer 1

Consider the special case where $A$ is measurable. Then the right side of your inequality is $\lambda(A)-\lambda^*(B)$, which is the inner measure of $B$. So, unless $B$ is also measurable, your proposed inequality won't hold: Outer measure is $\leq$ inner measure only for measurable sets.