- Let E be a Lebesgue measurable set in ℝ. Suppose m(E) > 0. Consider the set
Show that E − E contains an interval. Hint: Let

Note f is continuous at 0 and f(0) > 0 and use continuity of translation in L

^{p}. - Establish the inequality
_{r}≤_{p}_{q}whenever=+. - Let be counting measure on ℕ. Thus Ω = ℕ and S = Pwith μ= number of things in S. Let 1 ≤ p ≤ q. Show that in this case,
Hint: This is real easy if you consider what ∫

_{Ω}fdμ equals. How are the norms related? - Consider the function, f=+for x,y > 0 and+= 1. Show directly that f≥ 1 for all such x,y and show this implies xy ≤+.
- Give an example of a sequence of functions in L
^{p}which converges to zero in L^{p}but does not converge pointwise to 0. Does this contradict the proof of the theorem that L^{p}is complete? - Let K be a bounded subset of L
^{p}and suppose that there exists G such that G is compact withand for all ε > 0, there exist a δ > 0 and such that if

< δ, thenfor all u ∈ K. Show that K is precompact in L

^{p}. Hint: Let ϕ_{k}be a mollifier and considerVerify the conditions of the Ascoli Arzela theorem 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 ℝ

^{n}? - Let be a metric space and suppose also thatis a regular measure space such that μ< ∞ and let f ∈ L
^{1}where f has complex values. Show that for every ε > 0, there exists an open set of measure less than ε, denoted here by V and a continuous function, g defined on Ω such that f = g on V^{C}. Thus, aside from a set of small measure, f is continuous. If≤ M, show that it can be assumed that≤ M. This is called Lusin’s theorem. Hint: Use Theorems 22.2.4 and 22.1.10 to obtain a sequence of functions in C_{c},which converges pointwise a.e. to f and then use Egoroff’s theorem to obtain a small set, W of measure less than ε∕2 such that convergence is uniform on W^{C}. Now let F be a closed subset of W^{C}such that μ< ε∕2. Let V = F^{C}. Thus μ< ε and on F = V^{C}, the convergence ofis uniform showing that the restriction of f to V^{C}is continuous. Now use the Tietze extension theorem. - Let ϕ
_{m}∈ C_{c}^{∞}(ℝ^{n}),ϕ_{m}≥ 0,and ∫_{ℝn}ϕ_{m}(y)dy = 1 withShow if f ∈ L

^{p}(ℝ^{n}),lim_{m→∞}f ∗ ϕ_{m}= f in L^{p}(ℝ^{n}). - Let ϕ : ℝ → ℝ be convex. This means
whenever λ ∈ [0,1]. Verify that if x < y < z, then

≤and that≤. Show if s ∈ ℝ there exists λ such that ϕ(s) ≤ ϕ(t) + λ(s−t) for all t. Show that if ϕ is convex, then ϕ is continuous. - ↑ Prove Jensen’s inequality. If ϕ : ℝ → ℝ is convex, μ(Ω) = 1, and f : Ω → ℝ is in L
^{1}(Ω), then ϕ(∫_{Ω}fdu) ≤∫_{Ω}ϕ(f)dμ. Hint: Let s = ∫_{Ω}fdμ and use Problem 9. - Let += 1 ,p > 1, let f ∈ L
^{p}(ℝ),g ∈ L^{p′ }(ℝ). Show f ∗ g is uniformly continuous on ℝ and |(f ∗g)(x)|≤||f||_{Lp}||g||_{Lp′}. Hint: You need to consider why f ∗g exists and then this follows from the definition of convolution and continuity of translation in L^{p}. - B(p,q) = ∫
_{0}^{1}x^{p−1}(1 − x)^{q−1}dx,Γ(p) = ∫_{0}^{∞}e^{−t}t^{p−1}dt for p,q > 0. The first of these is called the beta function, while the second is the gamma function. Show a.)Γ(p + 1) = pΓ(p);b.) Γ(p)Γ(q) = B(p,q)Γ(p + q). - Let f ∈ C
_{c}(0,∞) and define F(x) =∫_{0}^{x}f(t)dt. ShowHint: Argue there is no loss of generality in assuming f ≥ 0 and then assume this is so. Integrate ∫

_{0}^{∞}|F(x)|^{p}dx by parts as follows:Now show xF

^{′}= f − F and use this in the last integral. Complete the argument by using Holder’s inequality and p − 1 = p∕q. - ↑ Now suppose f ∈ L
^{p}(0,∞),p > 1, and f not necessarily in C_{c}(0,∞). Show that F(x) =∫_{0}^{x}f(t)dt still makes sense for each x > 0. Show the inequality of Problem 13 is still valid. This inequality is called Hardy’s inequality. Hint: To show this, use the above inequality along with the density of C_{c}in L^{p}. - Suppose f,g ≥ 0. When does equality hold in Holder’s inequality?
- ↑ Show the Vitali Convergence theorem implies the Dominated Convergence theorem for finite measure spaces but there exist examples where the Vitali convergence theorem works and the dominated convergence theorem does not.
- ↑ Suppose μ(Ω) < ∞,{f
_{n}}⊆ L^{1}(Ω), andfor all n where h is a continuous, nonnegative function satisfying

Show {f

_{n}} is uniformly integrable. In applications, this often occurs in the form of a bound on ||f_{n}||_{p}. - ↑ Sometimes, especially in books on probability, a different definition of uniform integrability is
used than that presented here. A set of functions, S, defined on a finite measure space,
is said to be uniformly integrable if for all ε > 0 there exists α > 0 such that for all f ∈S,
Show that this definition is equivalent to the definition of uniform integrability with the addition of the condition that there is a constant, C < ∞ such that

for all f ∈S.

- f ∈ L
^{∞}(Ω,μ) if there exists a set of measure zero, E, and a constant C < ∞ such that |f(x)|≤ C for all xE.Show ||||

_{∞}is a norm on L^{∞}(Ω,μ) provided f and g are identified if f(x) = g(x)a.e. Show L^{∞}(Ω,μ) is complete. Hint: You might want to show thathas measure zero so_{∞}is the smallest number at least as large asfor a.e. x. Thus_{∞}is one of the constants, C in the above. - Suppose f ∈ L
^{∞}∩ L^{1}. Show lim_{p→∞}||f||_{Lp}= ||f||_{∞}. Hint:Now raise both ends to the 1∕p power and take liminf and limsup as p →∞. You should get

_{∞}− ε ≤ liminf_{p}≤ limsup_{p}≤_{∞} - Suppose μ(Ω) < ∞. Show that if 1 ≤ p < q, then L
^{q}(Ω) ⊆ L^{p}(Ω). Hint Use Holder’s inequality. - Show L
^{1}(ℝ) ⊈ L^{2}(ℝ) and L^{2}(ℝ) ⊈ L^{1}(ℝ) if Lebesgue measure is used. Hint: Consider 1∕and 1 ∕x. - Suppose that θ ∈ [0,1] and r,s,q > 0 with
show that

If q,r,s ≥ 1 this says that

Using this, show that

Hint:

Now note that 1 =

+and use Holder’s inequality. - Suppose f is a function in L
^{1}and f is infinitely differentiable. Is f^{′}∈ L^{1}? Hint: What if ϕ ∈ C_{c}^{∞}and f= ϕfor x ∈, f= 0 if x < 0? - Let be a measure space with μ a Radon measure. That is, it is regular, ℱ contains the Borel sets, μ is complete, and finite on compact sets. Let A be a measurable set. Show that for a.e.x ∈ A,
Such points are called “points of density”. Hint: The above quotient is nothing more than

Now consider Corollary 22.6.3.

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