Utrecht University, blok 4, 2019/ 2020 F. Ziltener WISB212, Analyse in meer variabelen Assignment 6 The exercises marked with a ∗ are inleveropdrachten.
The exercises marked with a + are particularly important.
The following exercise was used in the proof of the characterization of Riemann integrability in terms of the set of discontinuity.
+ Exercise 1 (countable union of sets of 0 Lebesgue measure) Prove that every countable uni- on of sets of 0 Lebesgue measure has 0 Lebesgue measure.
* Exercise 2 (two-dimensional integrals) (i) Prove that the function R 2 f : [0, 1] × [0, 1] → , f(x) := x1x2, is properly Riemann integrable and calculate its integral.
(ii) Prove that the function √ f : [0, 1] × [1, 4] → R, f(x) := ex1 x2 , is properly Riemann integrable and calculate its integral. (iii) Calculate Z 1 Z 1 x1 arctan (e x2) dx1 dx2. −1 0
Exercise 3 (tensor product) Let Q ⊆ Rm and R ⊆ Rn be rectangles, and f : Q → R and g : R → R be (properly) Riemann-integrable functions. Show that the function
f ⊗ g : Q × R → R, f ⊗ g(y, z) := f(y)g(z) is Riemann-integrable and express its integral in terms of the integrals of f and g.
Remark: This function is called the tensor product of f and g.
n p Exercise 4 (graph negligible) Let n ∈ N0, p ∈ N, K ⊆ R be compact, and f : K → R be continuous. Show that the graph of f is a negligible subset of Rn+p.
+ Exercise 5 (rescaling and translation) (i) Let f ∈ RI(Rn), i.e., f is a properly Riemann- integrable function on Rn. (Every such function is bounded and vanishes outside some bounded set.) Let c ∈ (0, ∞) and v ∈ Rn. We define x − v fe : Rn → R, fe(x) := f e . e c
Show that fe ∈ RI(Rn) and Z Z n fe(xe)dxe = c f(x)dx. Rn Rn
1 (ii) Let A ⊆ Rn be a Jordan-measurable set, c ∈ [0, ∞), and v ∈ Rn. Prove that the set Ae := cA + v := cx + v x ∈ A is Jordan-measurable, with Jordan-measure
|cA + v| = |c|n|A|.
* Exercise 6 (volume of simplex (pyramid)) (i) Draw the set
n ∆n := x ∈ R x1, . . . , xn ≥ 0, x1 + ··· + xn ≤ 1 for n = 1, 2, 3.
(ii) Prove that ∆n is Jordan-measurable. (iii) Calculate its Jordan-measure.
Remark: This set is called the standard n-simplex.
The following result will be used to calculate the volume of a ball in Rn.
Exercise 7 (integral of power of cosine) Prove that for n ∈ N0 (n − 1)(n − 3) ··· 1 π π, if n is even, Z 2 cosn t dt = n(n − 2) ··· 2 − π (n − 1)(n − 3) ··· 2 2 · 2, if n is odd. n(n − 2) ··· 3 Here we use the convention that the empty product equals 1. This product occurs in the cases n = 0, 1.
2 + Exercise 8 (volume of ball) Let n ∈ N0. n n Rn (i) Prove that the closed unit ball B := B1 ⊆ is Jordan-measurable. (ii) Calculate its Jordan-measure.
Exercise 9 (Cantor set negligible) Prove that the Cantor set K is a negligible subset of R.
Hint: Recall that \ K := Kk,
k∈N0 where Kk is recursively defined by 1 1 2 K := [0, 1],K := K ∪ K + . 0 k+1 3 k 3 k 3
Remark: The Cantor set is uncountable. (Why?) Hence it is an example of an uncountable negligible subset of R.
R x± −x2 Exercise 10 (integral of Gaußfunction) Show that 0 e dx converges, as x± → ±∞, and
x+ 0 Z 2 Z 2 Z 2 e−x dx := lim e−x dx + lim e−x dx ≤ 3. x+→∞ x−→−∞ R 0 x−
3