8.8. EXERCISES 213

11. Show that the continuous functions with compact support are dense in L1 (Rp) withrespect to Lebesgue measure. Will this work for a general Radon measure? Hint:You should show that the simple functions are dense in L1 (Rp) using the norm inL1 (Rp) and then consider regularity of the measure.

12. Suppose A ⊆ Rp is covered by a finite collection of Balls, F . Show that then thereexists a disjoint collection of these balls, {Bi}m

i=1, such that A⊆∪mi=1B̂i where B̂i has

the same center as Bi but 3 times the radius. Hint: Since the collection of balls isfinite, they can be arranged in order of decreasing radius.

13. This problem will help to understand that a certain kind of function exists. f (x) ={e−1/x2

if x ̸= 00 if x = 0

show that f is infinitely differentiable. Note that you only need

to be concerned with what happens at 0. There is no question elsewhere. This is alittle fussy but is not too hard.

14. ↑Let f (x) be as given above. Now let f̂ (x) ≡{

f (x) if x≤ 00 if x > 0 . Show that f̂ (x)

is also infinitely differentiable. Let r > 0 and define g(x) ≡ f̂ (−(x− r)) f̂ (x+ r).Show that g is infinitely differentiable and vanishes for |x| ≥ r. Let

ψ (x) =n

∏k=1

g(xk)

For U = B(0,2r) with the norm given by ∥x∥ = max{|xk| ,k ≤ n} , show that ψ ∈C∞

c (U).

15. ↑Using the above problem, let ψ ∈ C∞c (B(0,1)) . Also let ψ ≥ 0 as in the above

problem. Show there exists ψ ≥ 0 such that ψ ∈C∞c (B(0,1)) and

∫ψdmn = 1. Now

define ψk (x)≡ knψ (kx) . Show that ψk equals zero off a compact subset of B(0, 1

k

)and

∫ψkdmn = 1. We say that spt(ψk) ⊆ B

(0, 1

k

). spt( f ) is defined as the closure

of the set on which f is not equal to 0. Such a sequence of functions as just defined{ψk} where

∫ψkdmn = 1 and ψk ≥ 0 and spt(ψk)⊆ B

(0, 1

k

)is called a mollifier.

16. ↑It is important to be able to approximate functions with those which are infinitelydifferentiable. Suppose f ∈ L1 (Rp) and let {ψk} be a mollifier as above. We definethe convolution as follows.

f ∗ψk (x)≡∫

f (x−y)ψk (y)dmn (y)

Here the notation means that the variable of integration is y. Show that f ∗ψk (x)exists and equals

∫ψk (x−y) f (y)dmn (y) . Now show using the dominated conver-

gence theorem that f ∗ψk is infinitely differentiable. Next show that

limk→∞

∫| f (x)− f ∗ψk (x)|dmn = 0

Thus, in terms of being close in L1 (Rp) , every function in L1 (Rp) is close to onewhich is infinitely differentiable.