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Sets of Measure Zero in RN 10. SETS OF MEASURE 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 set 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 interval (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 Lebesgue measure 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 function: 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 integration by parts, while the other two are proved by a direct evaluation of the integral. 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 space 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 derivatives 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 gradient 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 calculus 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 multivariable calculus that such a transformation in a plane or space is a change of variables: 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 image 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.
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