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In this course so far, we have focused primarily on one specific example of a countably additive measure, namely Lebesgue measure. This measure was constructed from a more primitive concept of Lebesgue outer measure, which in turn was constructed from the even more primitive concept of elementary measure.
It turns out that both of these constructions can be abstracted. In this set of notes, we will give the Carathéodory extension theorem, which constructs a countably additive measure from any abstract outer measure; this generalises the construction of Lebesgue measure from Lebesgue outer measure. One can in turn construct outer measures from another concept known as a pre-measure, of which elementary measure is a typical example.
With these tools, one can start constructing many more measures, such as Lebesgue-Stieltjes measures, product measures, and Hausdorff measures. With a little more effort, one can also establish the Kolmogorov extension theorem, which allows one to construct a variety of measures on infinite-dimensional spaces, and is of particular importance in the foundations of probability theory, as it allows one to set up probability spaces associated to both discrete and continuous random processes, even if they have infinite length.
The most important result about product measure, beyond the fact that it exists, is that one can use it to evaluate iterated integrals, and to interchange their order, provided that the integrand is either unsigned or absolutely integrable. This fact is known as the Fubini-Tonelli theorem, and is an absolutely indispensable tool for computing integrals, and for deducing higher-dimensional results from lower-dimensional ones.
This is the final set of notes in this sequence. If time permits, the course will then begin covering the 245B notes, starting with the material on signed measures and the Radon-Nikodym-Lebesgue theorem.
In this supplemental note to the previous lecture notes, I would like to give an alternate proof of a (weak form of the) Carathéodory extension theorem. This argument is restricted to the -finite case, and does not extend the measure to quite as large a -algebra as is provided by the standard proof of this theorem, but I find it conceptually clearer (in particular, hewing quite closely to Littlewood’s principles, and the general Lebesgue philosophy of treating sets of small measure as negligible), and suffices for many standard applications of this theorem, in particular the construction of Lebesgue measure.
Let us first state the precise statement of the theorem:
Theorem 1. (Weak Carathéodory extension theorem) Let be a Boolean algebra of subsets of a set X, and let be a function obeying the following three properties:
- (Pre-countable additivity) If are disjoint and such that also lies in , then .
- (-finiteness) X can be covered by at most countably many sets in , each of which has finite -measure.
Let be the -algebra generated by . Then can be uniquely extended to a countably additive measure on .
We will refer to sets in as elementary sets and sets in as measurable sets. A typical example is when X=[0,1] and is the collection of all sets that are unions of finitely many intervals; in this case, are the Borel-measurable sets.