A question regarding measurable sets

Question

Let $A$ and $B$ be two disjoint subsets of $\mathbb R$. Let $m^*$ denote the Lebesgue outer measure on $\mathbb R$. Given

P: $m^*(A\cup B)=m^*(A)+m^*(B)$ Q: Both $A$ and $B$ are measurable R: One of $A$ and $B$ is measurable

The options are as follows:

1. If $P$ is true then $Q$ is true
2. If $P$ is not true then $R$ is true
3. If $R$ is true then $P$ is not true
4. If $R$ is true then $P$ is true

I know that $Q$ implies $P$. But does $P$ imply $Q$? Does $R$ imply $P$? Any suggestion will be appreciated.

Answer 1

Suggestion on 4) and indirectly on 3)

Let it be that $A$ is Lebesgue-measurable.

That means exactly that for each $C\subseteq\mathbb R$: $$m^{\star}C=m^{\star}\left(C\cap A\right)+m^{\star}\left(C\cap A^{c}\right)$$

Now substitute $C=A\cup B$.