.. role:: premium

.. 1998 June 23
.. ============

Exam 5 (36--42)
=========================

1998 Jun
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**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.

.. _1998Jun_p1:

.. proof:prob::

   The :term:`limit supremum <upper limit>` and :term:`limit infimum <lower limit>` of a sequence of sets :math:`\{E_n\}` are defined to be 

   .. math:: \varlimsup_n E_n = ⋂_n \left(⋃_{k≥ n} E_k\right) \quad \text{ and } \quad \varliminf_n E_n = ⋃_n \left(⋂_{k≥ n} E_k\right).

   a. Prove that :math:`\varliminf_n E_n ⊆ \varlimsup_n E_n`.

   b. Given an example of a sequence of sets for which :math:`\varliminf_n E_n ≠ \varlimsup_n E_n`.

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

.. _1998Jun_p2:

.. proof:prob::

   Prove: If :math:`f` and :math:`g` are :term:`continuous <continuous function>` real-valued functions on :math:`[0,1]`, and :math:`f(x) = g(x)` for almost all :math:`x`, then :math:`f(x) = g(x)` for all :math:`x`.

.. _1998Jun_p3:

.. proof:prob::

   Let :math:`ℕ` denote the set of natural numbers and :math:`\{p_n\}` a summable sequence of nonnegative real numbers.  For :math:`E⊆ N`, define :math:`μ(E) = ∑_{n∈ E}p_n`.
   
   a. Show that :math:`μ` is a :term:`measure` on the subsets of :math:`ℕ`.
   
   b. Give a formula for :math:`∫_ℕ f \, dμ` where :math:`f` is a bounded real-valued function on :math:`ℕ`.

.. _1998Jun_p4:

.. proof:prob::

   Let :math:`(X, 𝔐, μ)` be a :term:`measure space` and assume :math:`\{f_n\}_{n=1}^∞` is a sequence of nonnegative measurable functions such that :math:`f_n → f_0` a.e. and :math:`∫_X f_n \, dμ → 0`.  Show that :math:`∫_X f_0 \, dμ = 0`.

.. _1998Jun_p5:

.. proof:prob::

   a. Prove that for :math:`p≥ r>1`, :math:`L_p(0,1) ⊆ L_r(0,1)`

   b. Show that :math:`L_2(0,∞) ⊈ L_1(0,∞)`.

   The measure is :term:`Lebesgue measure` in each case.

.. _1998Jun_p6:

.. proof:prob::

   Let :math:`(X, 𝔐, μ)` be a :term:`measure space`, :math:`f` an :term:`integrable` function, and :math:`E_n = \{x : |f(x)|≥ n\}`.
   
   Show that :math:`n μ(E_n) → 0`.
   
.. _1998Jun_p7:

.. proof:prob::

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

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     :premium:`Solution to` :numref:`Problem {number} <1998Jun_p1>`

  (coming soon)

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     :premium:`Solution to` :numref:`Problem {number} <1998Jun_p2>`

  (coming soon)

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     :premium:`Solution to` :numref:`Problem {number} <1998Jun_p3>`

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       :premium:`Solution to` :numref:`Problem {number} <1998Jun_p4>`

    (coming soon)
    
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       :premium:`Solution to` :numref:`Problem {number} <1998Jun_p5>`

    (coming soon)

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       :premium:`Solution to` :numref:`Problem {number} <1998Jun_p6>`

    (coming soon)

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       :premium:`Solution to` :numref:`Problem {number} <1998Jun_p7>`

    (coming soon)

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