# Integration and Differentiation

Chapter
Part of the Universitext book series (UTX)

## Abstract

Let (a1,b1;…;a k ,b k ] be an interval in ℝ k , open on the left and closed on the right. Precisely stated, we assume that aj < bj for j = 1,…,k and the interval consists now of all points (x1,…,x k ) in ℝ k such that aj < xj ≤ bj for j = 1,…,k. We shall call an interval of this kind a cell. For reasons of convenience, the empty set will also be called a cell. Observe now that the collection Γ of all cells is not empty and it has the property that if A and B belong to Γ, then AB belongs to Γ and A\B can be written as a finite disjoint union ⋃C n of cells. In the case that B = A or BA, all C n in the finite union ⋃C n are then equal to the empty set. In view of the mentioned properties of Γ the collection Γ is called a semiring of subsets of ℝ k . Note that the collection of all intervals that are closed on the left and open on the right is likewise a semiring. On the other hand, the collection of all open (closed) intervals is not a semiring because the boundaries of the intervals cause difficulties. For any cell A = (a1,b1;…, a k ,b k ] we call the product
$$\prod\nolimits_j^k {_{ = 1}({b_j} - {a_j})}$$
(1)
measure A, and we denote this number by µ(A). Furthermore, we define µ(ф) = 0. Of course, to say that µ(A) is the measure of A is a neutral terminology for what is called the length of A if k = 1, the area of A if k = 2 and the content or volume of A if k = 3. The measure is, therefore, a map from Γ into It ℝ having the following properties:
1. (i)

µ, is non-negative and µ(ф) = 0,

2. (ii)

µ is monotone, i.e., AB in Γ implies µ(A) ≤ µ(B),

3. (iii)
µ is σ-additive, i.e., A =
$$\bigcup\nolimits_1^\infty {{A_n}}$$
(2)
(with A , all AnΓ and all A n mutually disjoint) implies
$$\mu (A) = \sum\nolimits_1^\infty {\mu ({A_n})} .$$
(3)

## Keywords

Equivalence Class Step Function Measure Zero Summable Function Finite Measure
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