nLab Caratheodory construction

Idea

The Carathéodory construction is a fundamental method in measure theory used to construct a measure space from an outer measure. It provides the standard machinery to extend a pre-measure (defined on a simple structure like a ring of sets) to a complete measure on a sigma-algebra. It is most famously used to define the Lebesgue measure.

Carathéodory’s Criterion

Definition

A subset $E \subseteq X$ is said to be Carathéodory-measurable (or $\mu^*$-measurable) if for every “test set” $A \subseteq X$, the following equality holds:

$$\mu^*(A) = \mu^*(A \cap E) + \mu^*(A \cap E^c)$$

In practice, since subadditivity implies $\mu^*(A) \leq \mu^*(A \cap E) + \mu^*(A \cap E^c)$, one only needs to verify:

$$\mu^*(A) \geq \mu^*(A \cap E) + \mu^*(A \cap E^c)$$

Properties

Theorem

(Carathéodory’s Theorem) Let $\mu^*$ be an outer measure on $X$. The collection $\mathcal{M}$ of all $\mu^*$-measurable sets forms a sigma-algebra on $X$. Furthermore, the restriction $\mu = \mu^*|_{\mathcal{M}}$ is a complete measure.

Applications

• Lebesgue Measure: Starting with the volume of $n$-dimensional intervals as a pre-measure, the Carathéodory construction yields the Lebesgue measure on $\mathbb{R}^n$.
• Hausdorff Measure: Used in fractal geometry? to define measures on metric spaces.
• Hahn-Kolmogorov Extension: The construction is the core engine behind extending measures from a ring of sets? to the generated $\sigma$-algebra.

Related entries

• measure
• outer measure
• measurable set
• Hahn-Kolmogorov theorem?