Lebesgue Measure
The Lebesgue measure is a map that operates in Rn.
Contents
Description
The problem with the Lebesgue outer measure is demonstrated as follows: Consider the open interval (0,1) and two subsets of it, A and B. These are taken such that A ∪ B = (0,1) and A n B = ∅. An ideal measure would give μ(A ∪ B) = μ(A) + μ(B) = 1. The Lebesgue outer measure however will resolve as μ(A) + μ(B) > 1.
The solution is to redefine the Lebesgue outer measure on the sets that behave well. This collection of sets is a subset of the power set, but also a superset of the Borel set. In this circumstance, 'behave well' means satisfying the Carathéodory criterion. A set B is considered Lebesgue measurable or μ*-measurable if, for all possible sets A, μ*(A) = μ*(A n B) + μ*(A ∪ BC).
As stated, this collection of μ*-measurable sets is a superset of the Borel set. However the collection can be formed by 'completing' the Borel set, which is to say: identifying all of the sets in the Borel set with measure equal to zero, and then adding all subsets of that set (of course with measure zero).
It is not straightforward to design a set in Rn that does not meet the Carathéodory criterion, but it would be strictly incorrect to say that all sets in P(Rn) meet it. The axiom of choice must be assumed to construct such an unmeasurable set. Nonetheless, it is useful to work on sets that are smaller than the power sets. Therefore the above construction, based on completing Borel sets, is ideal.