Outer Measures, Hahn-Kolmogorov Extension Theorem and $ {\pi}$-$ {\lambda}$ Lemma

  Outer measure turns out to be an important construction not only appearing in measure theory, but also in probability theory. One approach of developing Lebesgue measure is via outer measure, with the advantage being intuitive, as shown in the following. Along the way of construction, we motivate the Caratheodory's definition of measurability, as is suggested in this MO question. This way of thinking is also apparent in Terry's notes, which is also the major reference for this article.

  The Caratheodory extension theorem allows one to construct a measure from an outer measure. This in turn leads to Hahn-Kolmogorov extension theorem, which is a powerful tool to generate examples of measures. Roughly speaking, it standardizes the model of extending from more elementary finite additive measures, such as the Jordan measure, to desirable countably additive measures, such as the Lebesgue measure. It is also a nice place to put forward the Dykin's $ {\pi}$-$ {\lambda}$ machinery, which follows Amir Dembo's notes in probability theory.

  

Real Analysis: Abstract Integration Theory

In analysis one often makes use of a sequence to study the limiting process in some space, and consequently countable set-theoretic operations are inevitable. We would like a space that is stable under such operations, thus follows the definition of a $ {\sigma}$-algebra :

Definition 1 A $ {\sigma}$-algebra on a set $ {X}$ is a subset $ {\mathfrak{M}}$ of $ {\mathcal{P}(X)}$ that satisfies

1. $ {X\in\mathfrak{M}}$;
2. If $ {A\in\mathfrak{M}}$, then $ {A^{c}=\left(X-A\right)\in\mathfrak{M}}$;
3. If $ {\{A_{i}\}_{i=1}^{\infty}}$ is a countable family of elements in $ {A}$, then $ {\cup_{i}A_{i}\in\mathfrak{M}}$.

Elements of $ {\mathfrak{M}}$ are called measurable sets.

Here the letter "$ {\sigma}$'' refers to the German word "summe'', an abbreviation for "countable". Since $ {\cap_{i}A_{i}=\left(\cup_{i}A_{i}^{c}\right)^{c}}$, it follows that a $ {\sigma}$-algebra is also stable under countable intersections.

A set $ {X}$ with a $ {\sigma}$-algebra is called a measurable space. It is not necessarily a topological space, for instance in probability theory. Measure theory allows us to do integration on any set, provided on which a $ {\sigma}$-algebra is constructed.

However, from the Riemann integration theory one already notices a profound relation between the topological property of a function between topological spaces, viz. continuity and its Riemann integrability. Roughly speaking, over a compact set, the extent of discontinuity is the only obstruction to the Riemann integrability. For example, the characteristic function $ {\chi_{\mathbb{Q}\cap[0,1]}}$ of rational numbers in the unit interval is not Riemann integrable. But it is rather unsatisfactory, since the rational numbers is only a small "portion'' of the interval (it is countable), although wherein it is dense, in the sense that to each $ {p\in[0,1]}$ there is a sequence of rational numbers getting arbitrarily close.

In this note we will explore some general basic notions in abstract integration theory, following an axiomatic approach. This means we won't care about explicit constructions of $ {\sigma}$-algebra and measures at the moment. Construction of a $ {\sigma}$-algebra on a space will be discussed by a following chapter, also a more general notion of $ {L^{1}}$-integrability, as the completion of the notion of Riemann integrability. This completion is a special case in light of the unique continuous extension theorem in Banach spaces.