Characterization of (Lebesgue) measurable sets

Question

I am stuck trying to show a characterization of (Lebesgue) measurable sets from Caratheodory's definition of measurable. This means we do not have the notion of an inner measure, as it is not required to define measure via Caretheodory's criterion.

Show that a bounded subset $E\subset \mathbb{R}$ is (Lebesgue) measurable if there is some bounded interval $I\supset E$ such that $$\lambda^*(I)=\lambda^*(E)+\lambda^*(I\setminus E)$$

I am trying to prove that $E$ is measurable by showing that if such an $I$ exists, then for any interval $J$ $$\lambda^*(J)=\lambda^*(J\cap E)+\lambda^*(J\setminus E)$$

I am trying to prove this when $J\subset I$ (it is clearly true when $J\cap E=\emptyset$, and then the result will follow).

So I require $$\lambda^*(J)\ge\lambda^*(J\cap E)+\lambda^*(J\setminus E)$$ for $J\subset I$. Manipulating the set relations doesn't seem to get me anywhere.

Answer 1

This is one approach, thought I'm not sure its the one you're looking for:

Start by showing that $$\lambda_*(E)+\lambda^*(I\backslash E)=\lambda(I).$$ whenever $I$ is a bounded interval and $E$ is a subset of $I$ (I think it might be true for generally for $I$ measurable).

Then note that the hypothesis implies $\lambda_*(E)=\lambda^*(E)$.