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Math Help - Analysis, Dirichlet function

  1. #1
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    Analysis, Dirichlet function & Riemann function

    Denote by m the Lebesgue measure on X = (0, 1).

    1. Prove that the Dirichlet function d(\cdot) is not Riemann integrable.
    Since the Dirichlet function is discontinuous at every point x \in \mathbb{R} it is not Riemann integrable by definition.
    2. Prove that d(\cdot) is a non-negative Lebesgue measurable simple function.
    3. What is the Lebesgue integral \int d(x)dm?
    I think it is zero but I don't know how to show it.
    4. Prove that the Riemann function r(\cdot) is Riemann integrable.
    Last edited by JJMC89; April 27th 2010 at 02:18 PM. Reason: fixed 3
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  2. #2
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    For 2, ask yourself this: how might you define the Dirichlet function as the characteristic function of a set?

    For 3, I’m not quite sure what you mean by f_N(x).

    For 4, I am not sure if you know Lebesgue’s integrability criterion (this is a criterion for Riemann integrability due to Lebesgue, not a condition for Lebesgue integrability). Basically, a function is Riemann integrable iff it is continuous m-almost everywhere on the domain. So a straightforward way to show this is to note that the number of discontinuities of r is countable.
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  3. #3
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    Quote Originally Posted by Tikoloshe View Post
    For 3, I’m not quite sure what you mean by f_N(x).
    I fixed it. It should be d(x), not f_N(x)
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  4. #4
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    So what about the rest? Once you know 2, the answer of 3 follows as the measure of the set whose characteristic function you consider. If you are still having difficulty, try this:

    Write the definition of a characteristic function \chi_E(x) for some measurable set E. Then write the definition of d(x). Compare the two forms, and see if you find a set whose characteristic function is d.
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