Measure Theory and Integration

Abstract

This chapter provides sufficient general information about measures and integration for the purposes of this book. The starting point is the concept of an outer measure, which “measures weights of subsets of a space”. We should first consider how to sum such weights, which are either infinite or non-negative real numbers. For a finite set K, notation

$$ \sum\limits_{j \in K} {a_j } $$

abbreviates the usual sum of numbers a _j ∈ [0, ∞] over the index set K. The conventions here are that a+∞=∞ for all a ∈ [0, ∞], and that

$$ \sum\limits_{j \in 0/} {a_j = 0.} $$

Infinite summations are defined by limits as follows: Definition C.0.1. The sum of numbers a _j ∈ [0, ∞] over the index set J is

$$ \sum\limits_{j \in J} {a_j : = } \sup \left\{ {\sum\limits_{j \in K} {a_j :} K \subset J is finite} \right\}. $$

Exercise C.0.2. Let 0 < a _j < ∞ for each j ∈ J. Suppose

$$ \sum\limits_{j \in J} {a_j } < \infty . $$

Show that J is at most countable.