Our goal in this final chapter is to extend the notions of measure and integral to general sets. As an application, we shall include a brief discussion of some basic facts in probability theory. We saw in section 11.6 that Lebesgue measure, λ, can be defined by first introducing the Lebesgue outer measure, λ* (Definition 11.6.1), which is defined on P(ℝ) and then restricting it by means of Carathéodory’s definition (Theorem 11.6.1). As was pointed out there, this construction has the advantage that it can be carried out in general sets and this is what we intend to do here. Most of the results on Lebesgue measure and integral will therefore be extended and, since the proofs are in many cases almost identical, we may omit such proofs and assign them as exercises for the reader. Throughout this chapter, X will denote an arbitrary (nonempty) set. Also recall that, in the set [-∞, ∞] of extended real numbers, we have ±∞ · 0:= 0.
KeywordsMeasure Space General Measure Pairwise Disjoint Outer Measure Monotone Convergence Theorem
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