Chapter 14

The Lebesgue IntegralThis short chapter is on the Lebesgue integral. The emphasis is on the abstract Lebesgueintegral which is a general sort of construction depending on a measure space. It is anintroduction to this topic. I will use the generalized Riemann integral to give non trivialexamples in which the measure space is linked to the real line, thus tying it in to the topicof this book, advanced calculus for functions of one variable. Probably, these are the mostimportant examples. However, the complete development of this topic is in other sourceslike [25] or [17]. This is also in my on line analysis books. As mentioned earlier, this inte-gral can’t do some of the things the generalized Riemann integral can, but it is a lot easierto use if you are interested in things like function spaces or probability which are typicallybuilt on this integral. Also, you can consider absolute values of integrable functions andget functions for which the integral at least makes sense since this integral is free of someof the pathology associated with the generalized Riemann integral. The right way to do allof this is by the use of functionals defined on continuous functions which vanish off someinterval and to use the Riemann Stieltjes integrals, but to save trouble, I will emphasize themeasure of sets directly because the machinery of the generalized integral has been devel-oped. The example of measures on R is based on Dynkin’s lemma, a very useful result inprobability which is interesting for its own sake. This integral is like absolute convergentseries whereas the generalized Riemann integral is more like the inclusion of conditionallyconvergent series.

14.1 MeasuresThe definition of a measure is given next. It is a very general notion so I am presentingit in this way. The case of main interest here is where Ω = R. However, if you wantto study mathematical statistics or probability, it is very useful to understand this generalformulation. Surely the study of the integral should lead somewhere. It turns out that themachinery developed makes it very easy to extend to Lebesgue measure on appropriatesubsets of Rp also, but this will not be done in this book because this is a book on onevariable ideas. See my on line analysis books to see this done.

Definition 14.1.1 Let Ω be a nonempty set. A σ algebra F is a set whose elementsare subsets of Ω which satisfies the following.

1. If Ei ∈F , for i = 1,2, · · · , then ∪∞i=1Ei ∈F .

2. If E ∈F , then EC ≡Ω\E ∈F

3. /0,Ω are both in F

µ : F → [0,∞] is called a measure if whenever Ei ∈F and Ei ∩E j = /0 for all i ̸= j,then

µ (∪∞i=1Ei) =

∞

∑i=1

µ (Ei)

that sum is defined as supn ∑ni=1 µ (Ei) . It could be a real number or +∞. Such a pair

(Ω,F ) is called a measurable space. If you add in µ, written as (Ω,F ,µ) , it is called ameasure space.

321