Integration and Differentiation

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 a_j < b_j for j = 1,…,k and the interval consists now of all points (x₁,…,x _k ) in ℝ^k such that a_j < x_j ≤ b_j 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 A⋂B 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 B ⊃ A, 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 = (a_1,b_1;…, 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:

• (i) µ, is non-negative and µ(ф) = 0,

• (ii) µ is monotone, i.e., A ⊂ B in Γ implies µ(A) ≤ µ(B),

• (iii) µ is σ-additive, i.e., A =

  $$\bigcup\nolimits_1^\infty {{A_n}} $$

  (2)

  (with A , all A_n ∈ Γ and all A _n mutually disjoint) implies

  $$\mu (A) = \sum\nolimits_1^\infty {\mu ({A_n})} .$$

  (3)