Exam 5 (36–42)ΒΆ

1998 JunΒΆ

Instructions Attempt at least five of the following problems. Partial credit will be given for significant progress toward solutions. You may use standard results which have been established in courses, but indicate what result you are using when you use it.

Problem 36

The limit supremum and limit infimum of a sequence of sets \(\{E_n\}\) are defined to be

\[\varlimsup_n E_n = β‹‚_n \left(⋃_{kβ‰₯ n} E_k\right) \quad \text{ and } \quad \varliminf_n E_n = ⋃_n \left(β‹‚_{kβ‰₯ n} E_k\right).\]
  1. Prove that \(\varliminf_n E_n βŠ† \varlimsup_n E_n\).

  2. Given an example of a sequence of sets for which \(\varliminf_n E_n β‰  \varlimsup_n E_n\).

  3. If the sets \(E_n\) are measurable subsets of a measure space \((X, 𝔐, ΞΌ)\), make and prove the strongest statement possible about the relationship between \(m \left(\varliminf_n E_n\right)\) and \(\varliminf_n m(E_n)\).

Problem 37

Prove: If \(f\) and \(g\) are continuous real-valued functions on \([0,1]\), and \(f(x) = g(x)\) for almost all \(x\), then \(f(x) = g(x)\) for all \(x\).

Problem 38

Let \(β„•\) denote the set of natural numbers and \(\{p_n\}\) a summable sequence of nonnegative real numbers. For \(EβŠ† N\), define \(ΞΌ(E) = βˆ‘_{n∈ E}p_n\).

  1. Show that \(ΞΌ\) is a measure on the subsets of \(β„•\).

  2. Give a formula for \(∫_β„• f \, dΞΌ\) where \(f\) is a bounded real-valued function on \(β„•\).

Problem 39

Let \((X, 𝔐, ΞΌ)\) be a measure space and assume \(\{f_n\}_{n=1}^∞\) is a sequence of nonnegative measurable functions such that \(f_n β†’ f_0\) a.e. and \(∫_X f_n \, dΞΌ β†’ 0\). Show that \(∫_X f_0 \, dΞΌ = 0\).

Problem 40
  1. Prove that for \(pβ‰₯ r>1\), \(L_p(0,1) βŠ† L_r(0,1)\)

  2. Show that \(L_2(0,∞) ⊈ L_1(0,∞)\).

The measure is Lebesgue measure in each case.

Problem 41

Let \((X, 𝔐, ΞΌ)\) be a measure space, \(f\) an integrable function, and \(E_n = \{x : |f(x)|β‰₯ n\}\).

Show that \(n ΞΌ(E_n) β†’ 0\).

Problem 42

Find the two-dimensional (area) measure of the set \(\{(x,y) : 0≀ x≀ 1, 0≀ y≀ 1, |\sin x| < 1/2, \cos(x+y) \text{ is irrational}\}\).


Solution to Problem 36

(coming soon)

Solution to Problem 37

(coming soon)

Solution to Problem 38

(coming soon)

Solution to Problem 39

(coming soon)

Solution to Problem 40

(coming soon)

Solution to Problem 41

(coming soon)

Solution to Problem 42

(coming soon)


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Complex Analysis Exams

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