Outer Measure - Formal Definitions

Formal Definitions

An outer measure on a set X is a function

defined on all subsets of X, that satisfies the following conditions:

• Null empty set: The empty set has zero outer measure (see also: measure zero).

• Monotonicity: For any two subsets A and B of X,

• Countable subadditivity: For any sequence {A_j} of subsets of X (pairwise disjoint or not),

This allows us to define the concept of measurability as follows: a subset E of X is φ-measurable (or Carathéodory-measurable by φ) iff for every subset A of X

Theorem. The φ-measurable sets form a σ-algebra and φ restricted to the measurable sets is a countably additive complete measure.

For a proof of this theorem see the Halmos reference, section 11.

This method is known as the Carathéodory construction and is one way of arriving at the concept of Lebesgue measure that is so important for measure theory and the theory of integrals.