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Okay so the question is as follows:

Suppose that \(\displaystyle A \cup B\) is measurable and that

\(\displaystyle \lambda(A \cup B) = \lambda^{*}(A) + \lambda^{*}(B) < \infty\)

Prove that \(\displaystyle A\) and \(\displaystyle B\) are measurable.

So I know that \(\displaystyle \lambda^{*}(A) < \infty\) and \(\displaystyle \lambda^{*}(B) < \infty\) otherwise it would contradict \(\displaystyle \lambda(A \cup B) = \lambda^{*}(A) + \lambda^{*}(B) < \infty\) (since if one or both was infinity the sum would be infinity)

So I think it would be enough to show that inner measure is equal to the outer measure. We know that \(\displaystyle \lambda_{*}(A) \leq \lambda^{*}(A)\) for all sets \(\displaystyle A\) so I would need to show \(\displaystyle \lambda^{*}(A) \leq \lambda_{*}(A)\) but I'm not sure how to go about showing this. So first off am I on the right track? Second is how would I go about showing \(\displaystyle \lambda^{*}(A) \leq \lambda_{*}(A)\) if I am on the right track.

Any help is appreciated.