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Looking for Problem

  1. Jun 14, 2006 #1
    I'm looking for a good analysis problem (solved or otherwise) that will keep me busy for, say, a week. Anything up to Lebesgue integration is ok.
  2. jcsd
  3. Jun 14, 2006 #2
    It's not a very interesting problem, but it will require a bit of paper.

    Consider the following 11 subsets of the real numbers (let a and b be any real number):

    [tex] E_1 = \{(a,b)\} [/tex]
    [tex] E_2 = \{[a,b)\} [/tex]
    [tex] E_3 = \{(a,b]\} [/tex]
    [tex] E_4 = \{[a,b]\} [/tex]
    [tex] E_5 = \{(a,\infty)\} [/tex]
    [tex] E_6 = \{[a,\infty)\} [/tex]
    [tex] E_7 = \{(-\infty,b)\} [/tex]
    [tex] E_8 = \{(-\infty,b]\} [/tex]
    [tex] E_9 = \{all open sets\} [/tex]
    [tex] E_1_0 = \{all closed sets\} [/tex]
    [tex] E_1_1 = \{all compact sets\} [/tex]

    Show that the sigma-algebras generated by each of these sets is the same (that is, that they are all subsets of each other).
    Last edited: Jun 14, 2006
  4. Jun 14, 2006 #3
    Here's another one I like. Technically it's a calculus question, but you can make it Analysis by justifying each of the steps involved. You'll need at least one major theorem.

    [tex] \lim_{n \rightarrow \infty} \int_{0}^{\infty} e^{-2x} \sum_{k=0}^{n} \frac{x^k}{k!} dx [/tex]
    Last edited: Jun 14, 2006
  5. Jun 14, 2006 #4
    Thanks, but I was thinking something a little more theoretical and less repetitive.
  6. Jun 14, 2006 #5
    Suppose [tex] (X, M, \mu) [/tex] is a measure space. Suppose [tex] A_1, A_2, [/tex]... are subsets of X. Let:

    [tex] A = \{ x \in X \mid x \in A_n [/tex] for all but finitely many [tex] n \geq 1 \}[/tex],

    [tex] B = \{ x \in X \mid x \in A_n [/tex] for infinitely many [tex] n \geq 1\}[/tex].

    Question 1: Which, if any, of the following are equal to A or B?

    (a) [tex] \bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} A_n [/tex]

    (b) [tex] \bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} A_n [/tex]

    Question 2: Further suppose that [tex] A_1, A_2, ... \in M [/tex] and

    [tex] \sum_{n=1}^{\infty} \mu(A_n) < \infty [/tex]

    Prove that [tex] \mu(B) = 0 [/tex].
    Last edited: Jun 14, 2006
  7. Jun 14, 2006 #6
    Give an example of a ring of subsets of a set X that is not a sigma-ring.
  8. Jun 15, 2006 #7
    Suppose f:[a,b]->R is bounded and continuous except a finite number of points z_1, ..., z_k. Prove f:[a,b]->R is riemann integrable. (I believe this is also true if f is continuous except at a countable number of points)

    I'm not sure my proof is correct but I'd love to see yours if you come up with one so we can compare.
  9. Jun 15, 2006 #8
    Actually, it is true exactly when the discontinuities form a set of measure zero.
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