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Berkeley MATH 105 - Notes on Riemann Integral

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Notes on Riemann IntegralAn annex to H104 etc.Mariusz WodzickiDecember 2, 20101 Cells1.1 Intervals1.1.1 Connected subsets of RDefinition 1.1 A connected subset I of the topological space R is called aninterval.Exercise 1 Show that any connected subset I ∈ R contains (a, b) where a =inf S and b = sup S. (Hint: prove that, for any s, t ∈ S, if s < t, then[s, t] ⊆ S.)1.1.2It follows from Exercise 1 that any interval is of the formha, bi (−∞ ≤ a ≤ b ≤ ∞) (1)where ‘h’ stands for either ‘[’ or ‘(’ and ‘i’ stands for either ‘]’ or ‘)’.Definition 1.2 We shall say that an interval I is(a) nondegenerate if a < b,(b) open if I = (a, b),(c) closed if I = [a, b] or (−∞, b], or [a, ∞), or i = R,(d) bounded if −∞ < a and b < ∞.11.1.3For any interval I = ha, bi we shall denote its closure by¯I and its interiorby˚I .1.1.4 BoundaryThe set ∂I ˜ I \˚I will be called the boundary of I . It consists of at mosttwo points, and ∂I = ∅ precisely when I is open.1.1.5 LengthThe length of I = ha, bi will be denoted |I|˜ b − a.1.1.6For an interval I = ha, bi and δ > 0, we shall denote by Iδthe δ-neighborhood of I :Iδ˜ (a −δ, b + δ) (2)1.2 Cells1.2.1Definition 1.3 An n-cell is the Cartesian product of n intervalsI ˜ I1×···× In(3)It is naturally a subset of metric space Rn. A 1-cell is the same as an interval.Definition 1.4 We shall say that an n-cell is(a) nondegenerate if each Ijis nondegenerate,(b) open if each Ijis open,(c) closed if each Ijis closed,(d) bounded if each Ijis bounded.21.2.2A nonempty degenerate n-cell is isometric, as a metric space, to an m-cell for m = n − l where l is the number of factors in (3) which aredegenerate.1.2.3For any cell I we shall denote its closure by¯I and its interior by˚I.Exercise 2 Show that the intersection I ∩I0of two n-cells is again an n-cell—possibly degenerate or empty.1.2.4 BoundaryThe set ∂I ˜ I \˚I will be called the boundary of I.Exercise 3 Let I = [a1, b1] × ··· × [an, bn] with aj< bj, j = 1, . . . , n. Showthat ∂I is the union of 2n degenerate cells, each isometric to an (n − 1)-cell.The latter are called the faces of I.1.2.5 VolumeThe n-dimensional volume of a bounded cell, (3), is defined askIk˜ |I1|···|In|. (4)It is greater than zero precisely when I is nondegenerate.1.2.6 δ-thickeningFor a cell I and δ > 0, we shall denote by Iδthe open cellIδ˜ (I1)δ×···×(In)δ. (5)It is the smallest cell containing the δ-neighborhood of I.The volume of Iδsatisfies the following obvious estimatekIδk ≤ kIk+ 2δ l= n−1∑l=0nl(2δ)ldiam I)l−1!. (6)In particular, by selecting δ sufficiently small, one can make kIδk asclose to kIk as desired.32 Riemann Integral2.1 Outer contents and measure2.1.1For a family I of bounded n-cells we define kI k askI k˜∑I∈IkIk. (7)The quantity defined in (7) makes sense for any, even uncountable,family provided we define the sum in (7) assup{kI0k | I0⊆ I is finite}In particular, the values that kI k can take belong to [0, ∞].Exercise 4 Let S be an arbitrary set. Suppose that, for a function f : S−→[0, ∞),supn∑s∈S0f (s) | S0⊆ S is finiteo< ∞.Prove that the setsupp f ˜ {s ∈ S | f (s) , 0} (8)is countable1(the set defined in (8) is called the support of f ).2.1.2 Cell coversLet A be a subset of Rn.Definition 2.1 A cell cover of A is a family I of bounded nondegenerate cellssuch thatSI ⊇ A.2.1.3 Outer contentsDefinition 2.2 The infimum over all finite cell covers of A,¯m(A) ˜ inf{kI k | I is a finite cell cover of A}. (9)will be called the outer contents of subset A ⊆ Rn. When A is not bounded, Acannot be covered by finitely many bounded cells. In this case, we set¯m(A) =∞.1By countable we mean in these notes any set that can be embedded into the set ofnatural numbers. In particular, finite sets are ‘countable’ according to this definition.42.1.4 Outer measureDefinition 2.3 The infimum over all countable cell covers of A,¯µ(A) ˜ inf{kI k | I is a countable cell cover of A}. (10)will be called the outer measure of subset A ⊆ Rn.2.1.5Note that, in view of Exercise 4, we could have defined¯µ(A) as theinfimum of kI k over all cell covers since kI k = ∞ for any uncountablecover.2.1.6It follows directly from the definition that for any subsets of Rn:¯µ(A) ≤¯m(A) (11)and¯m(A) ≤¯m(B) as well as¯µ(A) ≤¯µ(B) (12)whenever A ⊆ B.Exercise 5 Let A = Q ∩[a, b]. Show that¯µ(A) = 0 while¯m(A) = b − a.Exercise 6 Prove that¯m [A∈AA!≤∑A∈A¯m(A), (13)for any finite family A of subsets of Rn, and¯µ [A∈AA!≤∑A∈A¯µ(A), (14)for any countable family A of subsets of Rn.52.1.7It follows directly from inequalities (12) and (13) that¯m(A ∩ A0) =¯m(A) =¯m(A0) =¯m(A ∪ A0) (15)whenever¯m(A \ A0) =¯m(A0\ A) = 0.Indeed, one has¯m(A ∩ A0) ≤¯m(A) ≤¯m(A ∪ A0)≤¯m(A ∩ A0) +¯m(A \ A0) +¯m(A0\ A) =¯m(A ∩ A0).2.1.8Similarly,¯µ(A ∩ A0) =¯µ(A) =¯µ(A0) =¯µ(A ∪ A0) (16)whenever¯µ(A \ A0) =¯µ(A0\ A) = 0.Exercise 7 Prove that in the definition of¯m(A) one could consider exclusivelyopen (respectively, closed) cell covers:¯m(A) = inf{kI k | I is a finite open cell cover of A} (17)= inf{kI k | I is a finite closed cell cover of A} (18)and, similarly, for¯µ(A). (Hint: for a cell cover I consider the family of clo-sures, {¯I | I ∈ I }, and the family of δ-thickenings, {Iδ| I ∈ I }, forsufficientlly small δ > 0.)2.1.9 Removal of overlapsFor any finite family of closed cells I , one can decompose each I ∈ Iinto a union of finitely many closed subcells so that the distinct subcells,I and I0, do not overlap, i.e., if I ∩I0is either empty or a degenerate cell.Denote a family obtained this way by J . Since every cell I ∈ I isthe union of cells from J , one has[J =[I (19)6andkJ k ≤ kI k (20)since the volume of every cell J ∈ J contributes to kJ k only oncewhile to kI k it contributes as many times as there are cells I ∈ I whichcontain it.In particular, in the definition of¯m(A) one could replace arbitraryfinite covers by finite families of closed nonoverlapping cells.Exercise 8 Produce an example showing that one cannot do the same in the caseof¯µ(A): the latter is generally smaller than the infimum of kJ k over all closednonoverlapping covers.2.1.10 The case of closed bounded subsetsCompactness of bounded closed subsets of Rnimplies that that the outermeasure and the outer contents of such sets

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