262 CHAPTER 10. CHANGE OF VARIABLES
10.9 Exercises1. Explain why for each t > 0,x→ e−tx is a function in L1 (R) and
∫∞
0 e−txdx = 1t . Thus∫ R
0
sin(t)t
dt =∫ R
0
∫∞
0sin(t)e−txdxdt
Now explain why you can change the order of integration in the above iterated in-tegral. Then compute what you get. Next pass to a limit as R → ∞ and show∫
∞
0sin(t)
t dt = 12 π . This is a very important integral. Note that the thing on the left
is an improper integral. sin(t)/t is not Lebesgue integrable because it is not ab-solutely integrable. That is
∫∞
0
∣∣ sin tt
∣∣dm = ∞. It is important to understand that theLebesgue theory of integration only applies to nonnegative functions and those whichare absolutely integrable.
2. Polar coordinates are x = r cos(θ) , y = r sin(θ). These transformation equationsmap [0,2π)× [0,∞) ontoR2. It is the restriction to this set of the same transformationdefined on the open set (−1,2π)× (−1,∞) and so from Theorem 10.3.3, if g isLebesgue measurable and zero off [0,2π)× [0,∞),∫
R2#(x)g(x)dm2 =
∫ 2π
0
∫∞
0g(r cosθ ,r sinθ)rdrdθ
where #(x) is 1 except for a set of measure zero consisting of either θ = 0 or r = 0.This set, has its image also of measure zero. Hence one can simply write∫
R2g(x)dm2 =
∫ 2π
0
∫∞
0g(r cosθ ,r sinθ)rdrdθ
Similar considerations apply to the general case of spherical coordinates as explainedabove. Use this change of variables for polar coordinates to show
∫∞
−∞e−x2
dx =√
π .
Hint: Let I =∫
∞
−∞e−x2
dx and explain why I2 =∫
∞
−∞
∫∞
−∞e−(x2+y2)dxdy. Now use
polar coordinates.
3. Let E be a Lebesgue measurable set in R. Suppose m(E)> 0. Consider the set
E−E = {x− y : x ∈ E,y ∈ E}.
Show that E−E contains an interval. Hint: Let f (x) =∫
XE(t)XE(x+ t)dt. Showf is continuous at 0 and f (0)> 0 and use continuity of translation in Lp.
4. Let K be a bounded subset of Lp (Rn) and suppose that there exists G such that G iscompact with
∫Rn\G |u(x)|
p dx < ε p and for all ε > 0, there exist a δ > 0 and suchthat if |h| < δ , then
∫|u(x+h)−u(x)|p dx < ε p for all u ∈ K. Show that K is pre-
compact in Lp (Rn). Hint: Let φ k be a mollifier and consider Kk ≡ {u∗φ k : u ∈ K} .The notation means the following:
u∗φ k (x)≡∫
u(x−y)φ k (y)dmn (y) =∫
u(y)φ k (x−y)dmn (y)
It is called the convolution. Verify the conditions of the Ascoli Arzela theorem Theo-rem 9.2.4 for these functions defined on G and show there is an ε net for each ε > 0.Can you modify this to let an arbitrary open set take the place of Rn?