Carathéodory’s criterion

• categories: Measure Theory, Definition

Carathéodory’s Criterion

Carathéodory’s criterion provides a condition for a set to be measurable with respect to the Lebesgue Measure. A set $E\subset {\mathbb{R}}^n$ is Lebesgue measurable if, for any set $A\subset {\mathbb{R}}^n$, the Outer Measure satisfies the following relation:

$$m^{\ast }(A)=m^{\ast }(A\cap E)+m^{\ast }(A\cap E^c),$$

where $m^{\ast }$ is the Lebesgue outer measure and $E^c$ is the complement of $E$.

Intuition:

The criterion essentially states that a set $E$ is measurable if its presence or absence does not distort the outer measure of any set $A$. In other words, $E$ “splits” any set $A$ cleanly into two parts, without causing irregularities in how the size (outer measure) of $A$ is computed.