Chapter 32

Measures and IntegralsIf you want to understand a decent theory of integration, you need to do something otherthan the Riemann integral. In particular, if probability is of interest, you must understandthe notion of measure theory and the abstract Lebesgue integral. It is also the case thatthese topics are much easier to follow than the extreme technicalities required for a rigorousdescription of the very inferior Riemann integral of a function of many variables. That iswhy I am placing this material in this elementary book. The rigorous description of theRiemann integral for functions of many variables is in my engineering math book and alsoin my earlier calculus book [21]. It is very technical and what you end up with is not nearlyas good. The usual solution to this problem is to simply leave out the rigorous presentationand pretend people understand it when they don’t. This is essentially what I did earlier inthe book and you will see this done even in advanced calculus courses. I attempted to makethe integral plausible through the use of iterated integrals. This required an emphasis onintegration over very simple regions, those for which you can actually compute the integral,and it avoids the fundamental questions.

There are two chapters devoted to this material. The first is on the abstract frameworkfor Lebesgue integration. It has a very different flavor than what you saw up till now. Thesecond chapter considers the special case of Lebesgue integration and measure in Rp. Ifyou understand the first of these chapters, this one will seem fairly easy. I believe it isworth mastering the abstract material in order to gain a more up to date understanding ofthe integral. However, this is only an introduction. I have neglected all the very importantmaterial on representation theorems and functions spaces and regularity of the measures.You can see this in my on line book Calculus of Real and Complex Variables which isintended to follow this book or in Real and Abstract Analysis also on my web page. Thereare many standard texts which also give this material such as [20, 27].

Notation 32.0.1 In this chapter Ω will be some nonempty set. It could be a subset of Rp,the integers, part of a probability space, a part of a manifold, etc. First of all, the notation[g < f ] is short for {ω ∈Ω : g(ω)< f (ω)} with other variants of this notation beingsimilar. Also, the convention, 0 ·∞ = 0 will be used to simplify the presentation wheneverit is convenient to do so. The notation a∧b means the minimum of a and b.

Also XE (ω) is defined as

XE (ω)≡{

1 if ω ∈ E0 if ω /∈ E

This is called the indicator function of the set E because it indicates whether ω ∈ E, 1 if ω

is in E and 0 if it is not.

32.1 Countable SetsThere are different kinds of infinity. The smallest one is called ℵ0 and is referred to ascountably infinite. The theory of the Lebesgue integral lives in the land of sequences andthe indices of these come from a countably infinite set. One considers countable inter-sections and unions. I will only include the minimum needed to understand measure andintegration. A much more complete treatment is in Hewitt and Stromberg [20].

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