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10. SETS OF ZERO AND RIEMANN INTEGRABILITY 71

10. Sets of measure zero and Riemann integrability 10.1. Sets of measure zero in RN . Definition 10.1. (Sets of measure zero in R) A of real numbers is said to have measure 0 if it can be covered by a union of open intervals of total length less than any preassigned positive number ε> 0. A point is evidently a set of measure zero. Indeed, a point can be covered by an (a,b) whose length b a can be made arbitrarily small. In particular, the open interval (a,b−), the closed interval [a,b], and semi-open intervals (a,b] and [a,b) differ from one another by at most two points, that is, by a set of measure zero. So, the length of all these intervals should be b a (should coincide with the length of (a,b)). − Sets of measure zero in RN are defined similarly. Definition 10.2. A set in RN is said to be of measure zero if it can be covered by a union of open balls whose total volume can be made smaller than any preassigned positive number. For brevity, the equation µ(Ω) = 0 will be used to say that a set Ω is a set of measure zero. This notation will be appreciated later when the concept of a is introduced.

The volume of a ball. The volume of B RN is a ⊂ πN/2 V (a)= aN = C aN N Γ(1 + N/2) N where Γ is Euler’s gamma : b Γ(z)= lim e−ttz−1 dt b→∞ Z0 It has the following properties: Γ(z +1) = zΓ(z) , Γ(1) = 1 , Γ(1/2) = √π The first one is established by , while the other two are proved by a direct evaluation of the . These properties can be used to compute VN for any integer N > 0. In particular, 2 2 2πa V1(a) = 2a , V2(a)= πa , V (a)= V −2(a) N N N 72 2. THE LEBESGUE INTEGRATION THEORY

Examples of sets of measure zero. A finite collection of points in is a set of measure zero. • A segment of a straight line of finite length L is a set of measure • zero. Indeed, let us split it into n pieces of length L/n. Each such segment can be covered by a ball of radius L/n centered at the midpoint of the segment. The total volume is V = nV (L/n)= C n(L/n)N 0 n N N → as n for any dimension N 2. So, the total volume can be made→∞ arbitrary small. ≥ Generalizing the previous example, a Euclidean space RM can • be viewed a hyper-plane in a higher dimensional Euclidean space RN , N > M. Any rectangular box in RM is a set of measure zero in RM . For example, a rectangle in a plane in a three-dimensional space is a set of measure zero. A proof of this assertion is left to the reader as an exercise. Any subset of a set of measure zero is also a set of measure • zero. Theorem 10.1. A countable union of sets of measure zero is also a set of measure zero. Let ∞

G = Gn . n[=1 and all Gn are sets of measure zero. Fix ε. Then Gn is contained in an open ball of volume ε/2n. Therefore G is contained in the union of such ball with the total volume being ∞ ε 1 1 2 1 3 ε 1 V = 2−nε = 1+ + + + = = ε 2 2 2 2 ··· 2 · 1 1 Xn=1       − 2 Since ε is arbitrary, G is a set of measure zero. The established theorem allows us to construct unbounded sets of measure zero. In particular, all rational numbers are countable. So, the rational numbers form a set of measure zero in R.

Remark. Are there sets of measure zero in R that are not countable? The answer is affirmative. There are uncountable collections of num- bers which contain no interval. One of the most famous examples is the Cantor set. 10. SETS OF MEASURE ZERO AND RIEMANN INTEGRABILITY 73

A line in space is also a set measure zero because it is a union of countably many line segments of a finite length. Similarly, any subspace RM of RN is a set of measure zero if M < N. What about curves and surfaces in space?

10.2. Smooth surfaces. A function g is said to be from the class Cp if it has continuous partial up to order p. A point set S is called a smooth surface in RN if in a neighborhood of any point, S is a level set of a function from the class C1 and its does not vanish ∂g ∂g ∂g g = , ,..., = 0 ∇ ∂x1 ∂x2 ∂xN  6 Recall that a level set is a collection of all points at which the function has a particular value (e.g., zero) S = x RN g(x) = 0 { ∈ | } Notations for partial derivatives. In what follows, partial derivatives are sometimes denotes as ∂g ∂g(x) ∂xg = , ∂jg = ∂x ∂xj for brevity.

A smooth surface in R3. Recall from multivariable that in three-dimensional space a smooth surface near each point is the graph z = f(x,y) of a two-variable function f whose partial derivatives are continuous functions. The vector ∂f ∂f n = , , 1 −∂x − ∂y  is a normal to the graph (a normal to a tangent plane to the graph). Note that owing the continuity of partial derivatives, the normal is continuous on the graph. This is a characteristic property of a smooth surface. A smooth surface cannot have sharp edges or corners at which the direction of n would change by jumps. Now consider level set g(x,y,z) = 0. Since the gradient g does not vanish, without loss of generality ∂g/∂z = 0 at some point∇ P of the level set. Then by the implicit function6 theorem the equation g = 0 can be solved in a neighborhood of P with respect to z, that is, there exists a function f(x,y) such that z = f(x,y) is a root of the equation. Moreover, the 74 2. THE LEBESGUE INTEGRATION THEORY function f is from the class C1 and

∂xg ∂yg ∂xf = , ∂yf = . −∂zg −∂zg The latter equations are known are implicit differentiation equations. Substituting them into the normal n, and pulling out the factor ∂zg, it is concluded that the gradient g is parallel to n. So, by the im- plicit function theorem, level sets of∇ a C1 function with non-vanishing gradient are smooth surfaces and the gradient is normal to them. This picture has a natural generalization to higher dimensional spaces to define a smooth surface in RN . The gradient g is normal to a level set of g, and changes continuously on it. ∇ Consider a transformation of a space which is a rule that assigns a unique point F (x) RN to every point x RN so that F (x) is from 1 ∈ ∈ the class C (all partial derivatives ∂Fj/∂xi are continuous) and its Jacobian does not vanish: ∂F F : RN RN , det j = 0 →  ∂xi  6 Recall from that such a transformation in a plane or space is a : yj = F (j(x), or a curvilinear coor- dinates. Any straight line in space is mapped by this transformation to a smooth curve (a curve that has a continuous unit tangent vector). Therefore any plane is mapped into a smooth surface. So, the question about measure of smooth surfaces can studied by investigating images of sets of measure zero under transformations from the class C1. Theorem 10.2. The of a set of measure zero Ω under a transformation F from the class C1 is a set of measure zero: µ(Ω) = 0 µ(F (Ω)) = 0 ⇒ A proof of this theorem can be found in1. For example, a transformation of R3 can be written in the form

x = F1(p,s,t) , y = F2(p,s,t) , z = F3(p,s,t) Then the coordinate plane p = 0 (or its portion) is a parametric surface

x = F1(0,s,t) , y = F2(0,s,t) , z = F3(0,s,t)

The partial derivatives ∂sF and ∂tF at p = 0 are vectors tangent to the surface. They cannot be zero and cannot be parallel because the Jacobian of the transformation is not zero. The cross product of these vectors is a vector normal to the surface. It changes continuously along

1J.M. Lee, Introduction to smooth manifolds 10. SETS OF MEASURE ZERO AND RIEMANN INTEGRABILITY 75 the surface (the surface is smooth). The reader is asked to verify the stated properties on the partial derivatives. Since a coordinate plane or any its portion is a set of measure zero in R3, any smooth surface in R3 is a set of measure zero. In particular, the transformation to spherical coordinates (r,φ,θ) defines parametric equations of a sphere of radius a as the image of a coordinate plane r = a: x = a sin(φ)cos(θ) , y = a sin(φ) sin(θ) , z = a cos(φ) where φ and θ are the zenith and polar angles, respectively. Similarly, one can obtain parametric equations of a cone and cylinder as the images of the coordinate planes φ = const and θ = const, respectively. So, a sphere is a set of measure zero. The reader is asked to calculate the cross product of vectors (∂φx,∂φy,∂φz) and (∂θx,∂θy,∂θz) to see that it is parallel to (x,y,z) (a normal to the sphere) at any point on the sphere. More generally, the image of any Ω RM RN , N > M, is a smooth M dimensional surface in RN and⊂ it is a⊂ set of measure zero in RN . The result− can also be formulated without embedding of RM into a higher dimensional space RN , N > M, and using a C1 transformation of RN . Any M dimensional surface in RN is a a continuous invertible deformation of−RM (or its part). Algebraically, such a deformation is defined by parametric equations of the surface embedded into RN . These equations define a continuous transformation F from RM (the space of parameters) into RN . Theorem 10.3. Let Ω be an open set in RM and the map F : Ω RN is from the class C1(Ω) and the the rank of the Jacobian matrix→ ∂Fj/∂xk, j = 1, 2,...,N, k = 1, 2,...,M, is equal to M < N. Then the image of Ω is a set of measure zero in RN , µ(F (Ω)) = 0

Remark. The condition that a transformation in Theorems 10.2 and 10.3 is from the class C1 is essential. If one takes a merely contin- uous transformation C0 (so that the image of a line is a curve, not necessarily smooth), then the theorems are false. There are so-called space-filling curves or surfaces. For example, if Ω = [0, 1] 0 R2 (a unit interval on the first coordinate axis), then one can×{ construct} ⊂ a continuous transformation F (t)=(x(t),y(t)) such that it maps this interval, t [0, 1], onto a square [0, 1] [0, 1]. Geometrically, it looks like a curve∈ filling the square (a set of× non-zero measure). In other 76 2. THE LEBESGUE INTEGRATION THEORY words, the parametric curve (x(t),y(t)) passes through every point of the square as t spans the interval2.

10.3. Boundary of a set. Given a set Ω, one can add to it all its limit points. The result is called the closure of Ω, denoted Ω¯ (recall the characteristic property of a limit point x is that any open ball centered at x contains points of Ω different from x). For example, the closure of B is the ball of radius a that also includes the sphere x = a. A point a | | x is an interior point of Ω if there is a ball Ba(x) (of a sufficiently small radius a) which is contained in Ω. For example, every point of an open ball is an interior point. Clearly, every interior point is a limit point, but the converse is false. Let Ωo be a collection of all interior points of Ω. Then the boundary ∂Ω of Ω is defined as the point set: ∂Ω= Ω¯ Ω \ o For example, let Ω = Ba. Then ∂Ω= B¯ B = x a x

10.4. Riemann integrable functions. Now the class of Riemann inte- grable functions can be described. Theorem 10.4. A function is Riemann integrable on a rectangular box in a Euclidean space if and only if it is not continuous at most on a set of measure zero Let Ω RN be bounded and closed and a function f be continuous ⊂ everywhere. Then χΩ(x)f(x) has jump discontinuities at the boundary

∂Ω. If the boundary is smooth or piecewise smooth, then χΩ (x)f(x)

2W. Rudin, Principles of , Exercises in Chapter 7 10. SETS OF MEASURE ZERO AND RIEMANN INTEGRABILITY 77 is not continuous on a set measure zero and, hence, f is Riemann integrable on Ω because

N N f(x) d x = χΩ(x)f(x) d x, Ω R ZΩ ZR ⊂ Another useful criterion to determine Riemann integrability reads: Corollary 10.1. Let Ω be closed and bounded and ∂Ω be piecewise smooth. If f is continuous on Ω except possibly on finitely many smooth surfaces, then f is Riemann integrable on Ω.

Additivity of the . Suppose that f is Riemann inte- grable on Ω. Let Ω be divided by a smooth (or piecewise smooth) surface into regions Ω1 and Ω2. Then

f(x) dN x = f(x) dN x + f(x) dN x ZΩ ZΩ1 ZΩ2

Continuity of the Riemann integral. Let f be Riemann integrable on Ω. Let Ωa Ω be a region obtained from Ω by removing points of all open balls of⊂ radius a centered at points of the boundary ∂Ω. Then

lim f(x) dN x = f(x) dN x →0+ a ZΩa ZΩ

10.5. Exercises.

1. Show that a plane in a there-dimensional space is a set of mea- sure zero.

2. Can an set in RN be a set of measure zero in RN if it has an interior point? Give an example or show that the answer is negative.