Lebesgue measurable implies caratheodory measurable The definition of Lebesgue measure is

$E$ is Lebesgue measurable if $\forall \epsilon > 0,$ there exists an open set $O$ such that $E \subset O$ and $m_*(O-E) < \epsilon$.

Caratheorody measurable is

$E$ is Caratheodory measurable, if $m_*(A) = m_*(A \cap E) + m_*(A \cap E^c)$ for all $A \in \Omega$.

I want to show that the top definition implies the bottom definition. Since $m_*(A) \le m_*(A \cap E) + m_*(A \cap E^c)$ is trivial by subadditivity, we only need to show $m_*(A) \ge m_*(A \cap E) + m_*(A \cap E^c)$. In order to use the Lebesgue measurability, let $E$ be Lebesgue measurable, and let $O$ be some open set such that $A \subset O$. Then
\begin{align*}
 m_*(A \cap E) + m_*(A \cap E^c) \le m_*(O \cap E) + m_*(O \cap E^c)
\end{align*}
But I'm not sure where I go from there. I want to use $\epsilon$ somewhere but I'm not sure how. Any hint would be much appreciated.
 A: Attention: We are going to use the terminology used in the question, including $m_*$ to denote Lebesgue outer measure:

$E$ is Lebesgue measurable if $\forall \epsilon > 0$, there exists an open set $O$ such that $E \subset O$ and $m_*(O-E) < \epsilon$.
and
$E$ is Caratheodory measurable, if $m_*(A) = m_*(A \cap E) + m_*(A \cap E^c)$ for all $A \subseteq \Omega$.

Let us prove that if $E$ is Lebesgue measurable, then $E$ is Caratheodory measurable.
Suppose $E$ is is Lebesgue measurable. Given any $\epsilon >0$, let $O_\epsilon$ be an open set such that  $E \subset O_\epsilon$ and $m_*(O_\epsilon-E) < \epsilon$.
Given any $A \subseteq \Omega$, we have, taking any  any $\epsilon >0$
\begin{align*}
\mu_*(A) &\leqslant \mu_*(A \cap E) + \mu(A\cap E^c) \\
& \leqslant \mu_*(A \cap O_\epsilon ) + \mu_*((A\cap O_\epsilon ^c) \cup (A\cap O_\epsilon \cap E^c))\\
&  \leqslant \mu_*(A \cap O_\epsilon ) + \mu_*(A\cap O_\epsilon ^c) +\mu_*(A\cap O_\epsilon \cap E^c)\\
& \leqslant \mu_*(A \cap O_\epsilon ) + \mu_*(A\cap O_\epsilon ^c) +\mu_*(O_\epsilon \cap E^c)\\
&  \leqslant \mu_*(A) + \epsilon
\end{align*}
where in the last step we used the fact that open sets are Caratheodory measurable.
So we have proved that, for any  $\epsilon >0$,
$$
\mu_*(A) \leqslant \mu_*(A \cap E) + \mu(A\cap E^c) \leqslant \mu_*(A) + \epsilon
$$
So
$$
\mu_*(A) = \mu_*(A \cap E) + \mu(A\cap E^c) 
$$
So $E$ is Caratheodory measurable.
