- Suppose A ⊆ ℝ
^{n}is covered by a finite collection of Balls, ℱ. Show that then there exists a disjoint collection of these balls,_{i=1}^{m}, such that A ⊆∪_{i=1}^{m}_{i}where_{i}has the same center as B_{i}but 3 times the radius. Hint: Since the collection of balls is finite, they can be arranged in order of decreasing radius. - In Problem 8, you showed that if f ∈ L
^{1}, there exists h which is continuous and equal to 0 off some compact set such thatDefine f

_{y}≡ f. Explain why f_{y}is Lebesgue measurable andNow justify the following formula.

is small enough. Explain continuity of translation in L^{1}which says that - This problem will help to understand that a certain kind of function exists.
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 a little fussy but is not too hard.

- ↑Let fbe as given above. Now let
Show that

is also infinitely differentiable. Now consider let r > 0 and define g≡. show that g is infinitely differentiable and vanishes for≥ r. Let ψ= ∏_{k=1}^{n}g. For U = Bwith the norm given by= max, show that ψ ∈ C_{c}^{∞}. - ↑Using the above problem, let ψ ∈ C
_{c}^{∞}. Also let ψ ≥ 0 as in the above problem. Show there exists ψ ≥ 0 such that ψ ∈ C_{c}^{∞}and ∫ ψdm_{n}= 1. Now defineShow that ψ

_{k}equals zero off a compact subset of Band ∫ ψ_{k}dm_{n}= 1. We say that spt⊆ B. sptis defined as the closure of the set on which f is not equal to 0. Such a sequence of functions as just definedwhere ∫ ψ_{k}dm_{n}= 1 and ψ_{k}≥ 0 and spt⊆ Bis called a mollifier. - ↑It is important to be able to approximate functions with those which are infinitely differentiable.
Suppose f ∈ L
^{1}and letbe a mollifier as above. We define the convolution as follows.Here the notation means that the variable of integration is y. Show that f ∗ ψ

_{k}exists and equalsNow show using the dominated convergence theorem that f ∗ψ

_{k}is infinitely differentiable. Next show thatThus, in terms of being close in L

^{1}, every function in L^{1}is close to one which is infinitely differentiable. - ↑From Problem 8 above and f ∈ L
^{1}, there exists h ∈ C_{c}, continuous and spta compact set, such thatNow consider h ∗ ψ

_{k}. Show that this function is in C_{c}^{∞}. The notation means you start with the compact set sptand fatten it up by adding the set B. It means x + y such that x ∈ sptand y ∈ B. Show the following. For all k large enough,so one can approximate with a function which is infinitely differentiable and also has compact support. Also show that h∗ψ

_{k}converges uniformly to h. If h is a function in C^{k}in addition to being continuous with compact support, show that for each≤ k,D^{α}→ D^{α}h uniformly. Hint: If you do this for a single partial derivative, you will see how it works in general. - ↑Let f ∈ L
^{1}. Show thatHint: Use the result of the above problem to obtain g ∈ C

_{c}^{∞}, continuous and zero off a compact set, such thatThen show that

You can do this by integration by parts. Then consider this.

This is the celebrated Riemann Lebesgue lemma which is the basis for all theorems about pointwise convergence of Fourier series.

- As another application of theory of regularity, here is a very important result. Suppose f ∈ L
^{1}and for every ψ ∈ C_{c}^{∞}Show that then it follows that f

= 0 for a.e.x. That is, there is a set of measure zero such that off this set f equals 0. Hint: What you can do is to let E be a measurable which is bounded and let K_{k}⊆ E ⊆ V_{k}where m_{n}< 2^{−k}. Here K_{k}is compact and V_{k}is open. By an earlier exercise, Problem 12 on Page 1457, there exists a function ϕ_{k}which is continuous, has values inequals 1 on K_{k}and spt⊆ V. To get this last part, show there exists W_{k}open such that_{k}⊆ V_{k}and W_{k}contains K_{k}. Then you use the problem to get spt⊆_{k}. Now you form η_{k}= ϕ_{k}∗ ψ_{l}whereis a mollifier. Show that for l large enough, η_{k}has values in,spt⊆ V_{k}and η_{k}∈ C_{c}^{∞}. Now explain why η_{k}→X_{E}off a set of measure zero. Then_{E}dm_{n}= 0 for every bounded measurable set E. Show that this implies that ∫ fX_{E}dm_{n}= 0 for every measurable E. Explain why this requires f = 0 a.e. The result which gets used over and over in all of this is the dominated convergence theorem. - This is from the advanced calculus book by Apostol. Justify the following argument using
convergence theorems.
Then you can take the derivative

and obtainNow you let x →∞. What happens to that first integral? It equals

and so it obviously converges to 0 as x →∞. Therefore, taking a limit yields

- The Dini derivates are as follows. In these formulas, f is a real valued function defined on ℝ.
Let V be an open set which contains N

_{ab}∩≡ N_{ab}^{r}such thatThen explain why there exist disjoint intervals

such thatand

each interval being contained in V ∩

. Thus you haveNext show there exist disjoint intervals

such that each of these is contained in some, theare disjoint, f−f≥ bm, and ∑_{j}m= m. Then you have the following thanks to the fact that f is increasing.showing that m

= 0 . This is for any r and so m= 0. Thus the derivative from the right exists for a.e. x by taking the complement of the union of the N_{ab}for a,b nonnegative rational numbers. Now do the same thing to show that the derivative from the left exists a.e. and finally, show that D_{−}f= D^{+}ffor almost a.e. x. Off the union of these three exceptional sets of measure zero all the derivates are the same and so the derivative of f exists a.e. In other words, an increasing function has a derivative a.e. - This problem is on Eggoroff’s theorem. Suppose you have a measure space where μ< ∞. Also suppose thatis a sequence of measurable, complex valued functions which converge to f pointwise. Then Eggoroff’s theorem says that for any ε > 0 there is a set N with μ< ε and convergence is uniform on N
^{C}.- Define E
_{mk}≡ ∪_{r=m}^{∞}. Show that E_{mk}⊇ E_{(m+1) k}for all m and that ∩_{m}E_{mk}= ∅ - Show that there exists msuch that μ< ε2
^{−k}. - Let N ≡∪
_{k=1}^{∞}E_{m(k) k}. Explain why μ< ε and that for all ωN^{C}, if r > m, then≤. Thus uniform convergence takes place on N^{C}.

- Define E
- Suppose you have a sequence which converges uniformly on each of sets A
_{1},,A_{n}. Why does the sequence converge uniformly on ∪_{i=1}^{n}A_{i}? - ↑Now suppose you have a where μ is a finite Radon measure and Ω is a metric space. For example, you could have Lebesgue measure for Ω a bounded open subset of ℝ
^{n}. Suppose you have f has nonnegative real values for all ω and is measurable. Then Lusin’s theorem says that for every ε > 0, there exists an open set V with measure less than ε and a continuous function defined on Ω such that f= gfor all ωV.- By Lemma 21.0.11, there exists an increasing sequence ⊆ C
_{c}which converges to f off a set N of measure zero. Use Eggoroff’s theorem to enlarge N tosuch that μ<and convergence is uniform off. - Next use outer regularity to obtain open V ⊇having measure less than ε. Thusconverges uniformly on V
^{C}. Therefore, that which it converges to is continuous on V^{C}a closed set. Now use the Tietze extension theorem.

- By Lemma 21.0.11, there exists an increasing sequence
- Let h : U → ℝ
^{n}be differentiable for U an open set in ℝ^{n}. Thus, as explained above, h takes measurable sets to measurable sets. Suppose h is one to one. For E a measurable subset of U define