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Math Help - content zero

  1. #1
    Senior Member Danneedshelp's Avatar
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    content zero

    Let f be bounded on [a,b]. Show that if the set of discontinuous points of f has content zero, then f is integrable.

    I am trying do a second semester analysis self study and I am pretty stuck on a few major concepts of integration. I would appriciate some help with this one. If any thing, I just need a place to get started.

    Here is what I have set up so far. I am given:

    D=\{d_{1}, d_{2},...,d_{N}\} has content zero and f is bounded on [a,b]. Thus, D\subseteq\bigcup_{n=1}^{N}O_{n} and \sum_{n=1}^{N}|O_{n}|\leq\epsilon (where, O_{n} represents open sets). Moreover, there exists an M>0 satistying |f(x)|\leq\\M for all x\in[a,b].

    My goal is to show there exists a partition P_{\epsilopn} such that U(f,P_{\epsilopn})-L(f,P_{\epsilopn})<\epsilon for any choice of
    \epsilon>0.

    Since the goal begins with a universal quantifier I let \epsilon>0 be arbitrary. Now, I need to show the existance of such a partition.

    As you can see I do not have much. I am not sure how to get started.

    Q1: Does the set of open intervals whos union contain the set D have to be nested?
    Q2: Can I assume [a,b] is compact to create a finite subcover and use that somehow?

    Thanks
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  2. #2
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    If I am reading this correctly the set D has only a finite number of elements in it. So try a easier problem first. What if f is bounded and is only discontinuous at one point c in [a,b]. How could you create a partition that that would show f is integrable. If you can solve this problem it will generalize to any finite set of points of discontinuity.

    Hint: squeeze the one point of discontinuity into a set of length \frac{\epsilon}{2M} then the function is continuous on [a,c-\frac{\epsilon}{4M}] and [c+\frac{\epsilon}{4M},b]
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  3. #3
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    The concept of content is a bit dated.
    If you are using the text material by T.H. Hildebrand then a subset E\subseteq [a,b] has content if and only if its characteristic function \chi_E is Riemann integrable on [a,b].
    That seems to be consistent with your notation. If so how can you use that?

    If that is not the definition of content you have, what textbook are you following?
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  4. #4
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by TheEmptySet View Post
    If I am reading this correctly the set D has only a finite number of elements in it. So try a easier problem first. What if f is bounded and is only discontinuous at one point c in [a,b]. How could you create a partition that that would show f is integrable. If you can solve this problem it will generalize to any finite set of points of discontinuity.

    Hint: squeeze the one point of discontinuity into a set of length \frac{\epsilon}{2M} then the function is continuous on [a,c-\frac{\epsilon}{4M}] and [c+\frac{\epsilon}{4M},b]
    Well, I would want to construct a partition that minimizes the effect of the discontinuity at c by embedding c in a very small subinterval.

    Quote Originally Posted by Plato View Post
    The concept of content is a bit dated.
    If you are using the text material by T.H. Hildebrand then a subset E\subseteq [a,b] has content if and only if its characteristic function \chi_E is Riemann integrable on [a,b].
    That seems to be consistent with your notation. If so how can you use that?

    If that is not the definition of content you have, what textbook are you following?
    I am using the book "Understanding Analysis" by Stephen Abbott. The book is current; the problem I posted is in a section called "integration functions with discontinuities".

    I have found very little on the subject in my google searchers. In the next section, "measure zero" is brought up and the only difference I can tell is that "measure zero" handles the infinitely countable case.
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  5. #5
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    Quote Originally Posted by Danneedshelp View Post
    Well, I would want to construct a partition that minimizes the effect of the discontinuity at c by embedding c in a very small subinterval.



    I am using the book "Understanding Analysis" by Stephen Abbott. The book is current; the problem I posted is in a section called "integration functions with discontinuities".

    I have found very little on the subject in my google searchers. In the next section, "measure zero" is brought up and the only difference I can tell is that "measure zero" handles the infinitely countable case.
    I do not know Abbots' text book.
    I your have reasonable university library access you may want to find Hildebrandt's book or a book by S K Breberian.
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  6. #6
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    A set X\subset \mathbb{R} has content zero iff for every \varepsilon >0 there exist (I_k)_{k=1}^{n} intervals with X \subset \cup I_k and \sum m(I_k) < \varepsilon.

    So in your problem, given \varepsilon >0 just use an argument completely analogous to that where you have a finite number of discontinuities.
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