Lebesgue Measure - Construction of The Lebesgue Measure

Construction of The Lebesgue Measure

The modern construction of the Lebesgue measure is an application of Carathéodory's extension theorem. It proceeds as follows.

Fix n ∈ N. A box in Rn is a set of the form

where b_i ≥ a_i, and the product symbol here represents a Cartesian product. The volume vol(B) of this box is defined to be

For any subset A of Rn, we can define its outer measure λ*(A) by:

We then define the set A to be Lebesgue measurable if for every subset S of Rn,

These Lebesgue measurable sets form a σ-algebra, and the Lebesgue measure is defined by λ(A) = λ*(A) for any Lebesgue measurable set A.

The existence of sets that are not Lebesgue measurable is a consequence of a certain set-theoretical axiom, the axiom of choice, which is independent from many of the conventional systems of axioms for set theory. The Vitali theorem, which follows from the axiom, states that there exist subsets of R that are not Lebesgue measurable. Assuming the axiom of choice, non-measurable sets with many surprising properties have been demonstrated, such as those of the Banach–Tarski paradox.

In 1970, Robert M. Solovay showed that the existence of sets that are not Lebesgue measurable is not provable within the framework of Zermelo–Fraenkel set theory in the absence of the axiom of choice (see Solovay's model).