ABSTRACT
It may well have occurred to the reader that there was a strong similarity to the procedures by which the Lebesgue and general Borel measures, and associated measure spaces, were constructed.
In each case we began with a rudimentary notion of “measure” as applied to a special collection of simple sets: (a) In the Lebesgue theory, the collection of simple sets was the collection of open intervals, and the measure of an interval (a, b) was defined as its standard interval length: https://www.w3.org/1998/Math/MathML"> | ( a , b ) | = b − a , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003257745/45ddd753-ce5a-45dd-beae-26c820031287/content/math125_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> whether the interval was finite or infinite. (b) In the Borel theory, the collection of simple sets was the collection of finite or infinite right semi-closed intervals, {(a, b]}, and the measure of a given interval was defined by its F-length: https://www.w3.org/1998/Math/MathML"> | ( a , b ) | F = F ( b ) − F ( a ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003257745/45ddd753-ce5a-45dd-beae-26c820031287/content/math125_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> again whether the interval was finite or infinite. Here F(x) was a given increasing, right continuous function.
In each case, this definition of interval measure was extended to a set function defined on all sets A of the power sigma algebra, σ(P(ℝ)). This extension was defined as the infimum of the total measures of all collections of simple sets which cover A. This was done in (2.4) and (5.9), respectively, and the set function extensions were called outer measures.
In the Lebesgue case, it was shown that Lebesgue outer measure could not be a Lebesgue measure due to the existence of a collection of highly irregular sets. Though not proved, this phenomenon is true in more general contexts. This implied that in order to eliminate such irregular sets, the power sigma algebra to which it was applied needed to be restricted.
A notion of measurable, which was formalized as Carathéodory measurable and named for Constantin Carathéodory (1873–1950), was introduced which provided such a restriction by requiring of measurable sets a certain type of regularity as defined in (2.14) and (5.10).
In both cases it was shown that the collection of Carathéodory measurable sets formed a sigma algebra which included the original collection of simple sets in item 1, as well as all sets with respective outer measure 0.
In both cases it was shown that when restricted to this sigma algebra, outer measure was a true measure, creating Lebesgue measure and general Borel measures, respectively. Further, these measures reproduced the values of the original measures on the collections of simple sets in item 1.
