Integration and Differentiation

Abstract

Let (a1,b1;…;a k ,b k ] be an interval in ℝk, 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 ℝk 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 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 ℝk. 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: (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 An ∈ Γ and all A n mutually disjoint) implies $$\mu (A) = \sum\nolimits_1^\infty {\mu ({A_n})} .$$ (3) KeywordsEquivalence ClassStep FunctionMeasure ZeroSummable FunctionFinite MeasureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.