Carathéodory's criterion

Remarks

The inequality $\le$ always holds by subadditivity, so only $\ge$ has content, and only for $A$ with $\lambda^*(A) < \infty$. If $E$ is Lebesgue measurable, approximate $E$ from outside by an open set, and use that open sets split every $A$ additively. Conversely, apply the splitting identity to bounded parts of $E$ and to open covers chosen close to $\lambda^*$. For example, a closed box $E$ splits every $A$ additively, in line with its being measurable. The criterion is useful because the splitting condition makes sense for any outer measure $\mu^*$ on any set $X$. For any outer measure, the sets $E$ with $\mu^*(A) = \mu^*(A \cap E) + \mu^*(A \setminus E)$ for all $A \subseteq X$ form a $\sigma$-algebra, and $\mu^*$ restricted to it is a complete measure.

Going deeper

For a student

In a measure theory course this splitting condition is often taken as the definition of a measurable set. With it, Carathéodory's construction turns an outer measure into a complete measure; this gives Lebesgue measure, Lebesgue–Stieltjes measures and Hausdorff measures. The same construction is the heart of the proof of the Carathéodory extension theorem. The theorem above shows that for $\R^n$ this abstract notion agrees with the geometric one of approximation by open sets.