Intervals and Outer Measure on $\mathbb{R}$

Abstract

This article gives some properties of intervals in $\mathbb{R}$ and discusses some problems involving intervals for which the concept of outer measure on $\mathbb{R}$ provides a more efficient solution than an elementary approach. The outer measure is then defined and some of its main properties in relation to intervals are developed, culminating in the countable additivity of outer measure on the 'system of intervals' $\mathcal{I} = \{$ all countable unions of intervals in $\mathbb{R}\ \}$. This demonstrates early on how the outer measure on $\mathbb{R}$ is naturally countably additive on a quite large class of sets, and motivates the Borel algebra $\mathcal{B}$ as an extension of that class which provides an additional desired property of outer measure, namely closure of its domain under set complementation -- for example as developed in [Axler, Chap 2]. Details are given of how one of the intervals problems solved by the outer measure allows proof prior to the Lebesgue integration theory of the Bounded Convergence, Monotone Convergence, and Dominated Convergence Theorems for Riemann integrals. One application of the latter is the proof of Stirling's Formula given in [Conrad]. Some further details on handling double series are provided than is normally given, based on the textbook 'Theory and Application of Infinite Series' by [Knopp] and the article [4]. The term 'countable union' of sets will mean a union of an infinite sequence of sets. $\overline{\mathbb{R}}$ will denote the extended real number system $\mathbb{R}\ \cup \{\infty, -\infty\}$.