PhD Qualify Exam, Analysis, Oct. 1, 1999 Show all works

PhD Qualify Exam, Analysis, Oct. 1, 1999

Show all works

1. (a)[5%] Please state the Radon-Nikodym Theorem.

(b)[5%] Please state the Fubini Theorem.

(c)[5%] Please define a Vitali Covering..

(d)[5%] Please state the Vitali Covering Theorem.

2.(a)[10%] Let f be a real-valued bounded measurable function on [a, b], take any r ∈ R, and define the function F : [a, b] → R by F (x) = r +^{∫ x}

a

f (t) dt. Prove that F⁰ = f a.e. and specify each place you invoke any form of the Domination Convergence Theorem.

(b)[10%] Let C ⊂ [0, 1] be the Cantor ternary set with associated Cantor ternary function fC

and associated Canter ternary measure µ_C (defined by : µ_C(a, b] = f_C(b)− fC(a) ). Prove that µ_C is a continuous measure, i.e., µ_C{x} = 0 for each x, and then show that U ∩ C is uncountable for every open set U for which U∩ C 6= φ.

3. (a)[10%] Given f ∈ L¹[0, 1]. Prove that for all ² > 0 there is a δ > 0 such that for every A∈ M, for which m(A) < δ, we can conclude that^∫

A|f(t)| dm(t) < ².

(b) Let X be a compact Hausdorff space and let C(X) be the real Banach space of all real-valued continuous functions on X with sup-norm. Prove the following:

(i) [5%] If L : C(X)→ R is a positive linear functional, then L is continuous.

(ii)[5%] If L : C(X)→ R is a continuous linear functional and {fn} ⊂ C(X) is a sup-norm bounded sequence which tends pointwise to a function f ∈ C(X), then lim L(fn) = L(f ).

4. (a)[10%] Assume the inequality

kfgk1 ≤ kfkpkgkq

holds for all functions f ∈ L^p and g∈ L^q. Show that the relation between p and q is 1 p +1

q = 1. ( Hint: Consider f_λ(x) = f (λx) and g_λ(x) = g(λx).)

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(b)[10%] Assume that f_n → f in measure. ( Definition of convergence in measure: for each

² > 0, there is an N > 0 such that µ{x : |fn(x)− f(x)| > ²} < ², for all n > N. ) Assume that

|fn| ≤ g, g ∈ L^p(µ), f ∈ L^p(µ), and µ is a σ-finite measure on X. Prove that f_n→ f in L^P(µ).

5. Suppose that x1, x2, x3, · · · is a sequence of points in the unit interval [0, 1] such that for every continuous real valued function f defined on [0, 1], lim1

n[f (x₁) +· · · + f(xn)] exists. Define this limit to be L(f ).

(a)[10%] Prove that there is a positive measure µ, defined on the σ-algebra of all Borel sets of [0, 1], such that

L(f ) =

∫

[0,1]

f dµ

for all continuous functions f . State any theorems you use.

(b)[10%] Prove that the measure µ in part (a) is Lebesgue measure if and only if for every integer k ≥ 1, L(x^k) = 1

k + 1, i.e., 1

n(x^k₁ +· · · + x^kn)→ 1

k + 1 as n→ ∞.

State any theorems you use.

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