Chapter 13

The Generalized Riemann IntegralThe preceding part of the book is essentially devoted to nineteenth century analysis. Thegeneralized Riemann integral is a relatively recent development from around 1957. How-ever, it is very close to the Riemann integral. One replaces the norm of the partition whichis a single number with a gauge function.

13.1 Definitions and Basic PropertiesThis chapter is on the generalized Riemann integral. The Riemann Darboux integral pre-sented earlier has been obsolete for over 100 years. The integral of this chapter is certainlynot obsolete and is in certain important ways the very best integral currently known. Thisintegral is called the generalized Riemann integral, also the Henstock Kurzweil integralafter the two people who invented it and sometimes the gauge integral. Other books whichdiscuss this integral are the books by Bartle [7], Bartle and Sherbert, [8], Henstock [16],or McLeod [22]. Considerably more is presented in some of these references. In whatfollows, F will be an increasing function, the most important example being F (x) = x. Inthe Stieltjes integral, we featured ∥P∥< δ . One does the same thing here except here δ isnot a positive number but a positive function.

Definition 13.1.1 Let [a,b] be a closed and bounded interval. A tagged division1

of [a,b] = I is a set of the form P ≡ {(Ii, ti)}ni=1 where ti ∈ Ii = [xi−1,xi], and a = xi−1 <

· · · < xn = b. Let the ti be referred to as the tags. A function δ : R→ (0,∞) is called agauge function or simply gauge for short. A tagged division P is called δ fine if

Ii ⊆ (ti−δ (ti) , ti +δ (ti)) .

A δ fine division, is understood to be tagged. More generally, a collection, {(Ii, ti)}pi=1 is δ

fine if the above inclusion holds for each of these intervals and their interiors are disjointeven if their union is not equal to the whole interval, [a,b]. In this definition, one oftenrequires that Ii ⊆ [ti−δ (ti) , ti +δ (ti)] rather than the open interval above. It appears tonot matter much in what is presented here.

The following fundamental result is essential.

Proposition 13.1.2 Let [a,b] be an interval and let δ be a gauge function on [a,b].Then there exists a δ fine tagged division of [a,b].

Proof: Suppose not. Then one of[a, a+b

2

]or[ a+b

2 ,b]

must fail to have a δ fine taggeddivision because if they both had such a δ fine division, the union of the two δ fine divi-sions would be a δ fine division of [a,b]. Denote by I1 the interval which does not have aδ fine division. Then repeat the above argument, dividing I1 into two equal intervals andpick the one, I2 which fails to have a δ fine division. Continue this way to get a nestedsequence of closed intervals, {Ii} having the property that each interval in the set fails tohave a δ fine division and diam(Ii)→ 0. Therefore, ∩∞

i=1Ii = {x} where x ∈ [a,b]. Now(x−δ (x) ,x+δ (x)) must contain some Ik because the diameters of these intervals con-verge to zero. It follows that {(Ik,x)} is a δ fine division of Ik, contrary to the constructionwhich required that none of these intervals had a δ fine division.

1In beginning calculus books, this is often called a partition and I followed this convention earlier. The word,division is a much better word to use. What the xi do is to “divide” the interval into little subintervals.

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