Exercise 1.7.2 (Compatibility with Lebesgue measurability)

We verify both directions of the equivalence.

• Suppose that $E$ is Carathéodory measurable. In particular, ${\mu }^{\text{∗}}(A)={\mu }^{\text{∗}}(A\cap E)+{\mu }^{\text{∗}}(A\setminus E)$ for any elementary set $A\subset {ℝ}^d$. Thus, by Exercise 1.2.17 (Carathéodory criterion, one direction), $E$ is Lebesgue measurable.

• Suppose that $E$ is Lebesgue measurable. By Definition 1.2.2 (Lebesgue measurability), fix an arbitrary $𝜖>0$ and let $O_{𝜖}$ be an open set such that $E\subset O_{𝜖}$ and $m^{\text{∗}}(O_{𝜖}-E)<𝜖$.
  Given any $A\subseteq \mathrm{\Omega }$, we have, taking any any $𝜖>0$

  $$\begin{array}{llll}\hfill {\mu }^{\text{∗}}(A)& \le {\mu }^{\text{∗}}(A\cap E)+{\mu }^{\text{∗}}(A\cap E^c)& \hfill & \\ \hfill & \le {\mu }^{\text{∗}}(A\cap O_{𝜖})+{\mu }^{\text{∗}}((A\cap O_{𝜖}^c)\cup (A\cap O_{𝜖}\cap E^c))& \hfill & \\ \hfill & \le {\mu }^{\text{∗}}(A\cap O_{𝜖})+{\mu }^{\text{∗}}(A\cap O_{𝜖}^c)+{\mu }^{\text{∗}}(A\cap O_{𝜖}\cap E^c)& \hfill & \\ \hfill & \le {\mu }^{\text{∗}}(A\cap O_{𝜖})+{\mu }^{\text{∗}}(A\cap O_{𝜖}^c)+{\mu }^{\text{∗}}(O_{𝜖}\cap E^c)& \hfill & \\ \hfill & \le {\mu }^{\text{∗}}(A)+𝜖& \hfill & \end{array}$$

  where in the last step we used the fact that open sets are Carathéodory measurable. To sum up, we have proved that, for any $𝜖>0$,

  $${\mu }_{\text{∗}}(A)\le {\mu }_{\text{∗}}(A\cap E)+\mu (A\cap E^c)\le {\mu }_{\ast }(A)+𝜖.$$

  Taking limits as $𝜖\to 0$, we obtain

  $${\mu }_{\text{∗}}(A)={\mu }_{\text{∗}}(A\cap E)+\mu (A\cap E^c).$$

  Thus, $E$ is Carathéodory measurable.