- Suppose A and B are sets of positive Lebesgue measure in ℝ
^{n}. Show that A − B must contain Bfor some c ∈ ℝ^{n}and ε > 0.Hint: First assume both sets are bounded. This creates no loss of generality. Next there exist a

_{0}∈ A, b_{0}∈ B and δ > 0 such thatNow explain why this implies

and

Explain why

Let

Explain why f

> 0. Next explain why f is continuous and why f> 0 for all x ∈ Bfor some ε > 0. Thus if< ε, there exists t such that x + t ∈ A−a_{0}and t ∈ B − b_{0}. Subtract these. - Show Mf is Borel measurable by verifying that ≡ E
_{λ}is actually an open set. Hint: If x ∈ E_{λ}then for some r,∫_{B(x,r) }dm > λm. Then for δ a small enough positive number, ∫_{B(x,r) }dm > λm. Now pick y ∈ Band argue that B⊇ B. Therefore show that,Thus B

⊆ E_{λ}. - Consider the following nested sequence of compact sets, {P
_{n}}.Let P_{1}=, P_{2}=∪, etc. To go from P_{n}to P_{n+1}, delete the open interval which is the middle third of each closed interval in P_{n}. Let P = ∩_{n=1}^{∞}P_{n}. By the finite intersection property of compact sets, P≠∅. Show m(P) = 0. If you feel ambitious also show there is a one to one onto mapping of [0,1] to P. The set P is called the Cantor set. Thus, although P has measure zero, it has the same number of points in it asin the sense that there is a one to one and onto mapping from one to the other. Hint: There are various ways of doing this last part but the most enlightenment is obtained by exploiting the topological properties of the Cantor set rather than some silly representation in terms of sums of powers of two and three. All you need to do is use the Schroder Bernstein theorem and show there is an onto map from the Cantor set to. If you do this right and remember the theorems about characterizations of compact metric spaces, Proposition 6.2.5 on Page 410, you may get a pretty good idea why every compact metric space is the continuous image of the Cantor set. - Consider the sequence of functions defined in the following way. Let f
_{1}= x on [0,1]. To get from f_{n}to f_{n+1}, let f_{n+1}= f_{n}on all intervals where f_{n}is constant. If f_{n}is nonconstant on [a,b], let f_{n+1}(a) = f_{n}(a),f_{n+1}(b) = f_{n}(b),f_{n+1}is piecewise linear and equal to( f_{n}(a) + f_{n}(b)) on the middle third of [a,b]. Sketch a few of these and you will see the pattern. The process of modifying a nonconstant section of the graph of this function is illustrated in the following picture.Show {f

_{n}} converges uniformly on [0,1]. If f(x) = lim_{n→∞}f_{n}(x), show that f(0) = 0,f(1) = 1,f is continuous, and f^{′}(x) = 0 for all xP where P is the Cantor set of Problem 3. This function is called the Cantor function.It is a very important example to remember. Note it has derivative equal to zero a.e. and yet it succeeds in climbing from 0 to 1. Explain why this interesting function is not absolutely continuous although it is continuous. Hint: This isn’t too hard if you focus on getting a careful estimate on the difference between two successive functions in the list considering only a typical small interval in which the change takes place. The above picture should be helpful. - A function, f : → ℝ is Lipschitz if≤ K. Show that every Lipschitz function is absolutely continuous. Thus every Lipschitz function is differentiable a.e., f
^{′}∈ L^{1}, and f− f= ∫_{x}^{y}f^{′}dt. - Suppose f,g are both absolutely continuous on . Show the product of these functions is also absolutely continuous. Explain why
^{′}= f^{′}g + g^{′}f and show the usual integration by parts formula - In Problem 4 f
^{′}failed to give the expected result for ∫_{a}^{b}f^{′}dx^{1}but at least f^{′}∈ L^{1}. Suppose f^{′}exists for f a continuous function defined on. Does it follow that f^{′}is measurable? Can you conclude f^{′}∈ L^{1}? - A sequence of sets, containing the point x is said to shrink to x nicely if there exists a sequence of positive numbers,and a positive constant, α such that r
_{i}→ 0 andShow the above theorems about differentiation of measures with respect to Lebesgue measure all have a version valid for E

_{i}replacing B. - Suppose F= ∫
_{a}^{x}fdt. Using the concept of nicely shrinking sets in Problem 8 show F^{′}= fa.e. - A random variable, X is a measurable real valued function defined on a measure
space, where P is just a measure with P= 1 called a probability measure. The distribution function for X is the function, F≡ Pin words, Fis the probability that X has values no larger than x. Show that F is a right continuous increasing function with the property that lim
_{x→−∞}F= 0 and lim_{x→∞}F= 1 . - Suppose F is an increasing right continuous function.
- Show that Lf ≡∫
_{a}^{b}fdF is a well defined positive linear functional on C_{c}where hereis a closed interval containing the support of f ∈ C_{c}. - Using the Riesz representation theorem for positive linear functionals
on C
_{c}, let μ denote the Radon measure determined by L. Show that μ= F− Fand μ= F− Fwhere F≡ lim_{x→b−}F. - Review Corollary 18.1.4 on Page 1829 at this point. Show that the
conditions of this corollary hold for μ and m. Consider μ
_{⊥}+ μ_{||}, the Lebesgue decomposition of μ where μ_{||}≪ m and there exists a set of m measure zero, N such that μ_{⊥}= μ_{⊥}. Show μ= μ_{⊥}+ ∫_{0}^{x}hdt for some h ∈ L^{1}. Using Theorem 24.7.7 show h= F^{′}m a.e. Explain why F= F+ S+ ∫_{0}^{x}F^{′}dt for some function, Swhich is increasing but has S^{′}= 0 a.e. Note this shows in particular that a right continuous increasing function has a derivative a.e.

- Show that Lf ≡∫
- Suppose now that G is just an increasing function defined on ℝ. Show that G
^{′}exists a.e. Hint: You can mimic the proof of Theorem 24.7.4. The Dini derivates are defined as_{+}G= D^{+}Gthe derivative from the right exists and when D^{−}G= D_{−}G, then the derivative from the left exists. Letbe an open interval and letLet V ⊆

be an open set containing N_{pq}such that m< m+ ε. Show using a Vitali covering theorem there is a disjoint sequence of intervals contained in V ,_{i=1}^{∞}such thatNext show there is a disjoint sequence of intervals

_{j=1}^{∞}such that each of these is contained in one of the former intervals andThen

= 0 . Taking a union of all N_{pq}for p,q rational, shows the derivative from the right exists a.e. Do a similar argument to show the derivative from the left exists a.e. and then show the derivative from the left equals the derivative from the right a.e. using a simlar argument. Thus G^{′}exists ona.e. and so it exists a.e. on ℝ becausewas arbitrary.

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