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Math Help - Measure Theory

  1. #1
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    Mar 2008
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    Unhappy Measure Theory

    Any help with any of these would be very much appreciated, even if it's just a link to some useful material, etc. Thanks in advance!

    1. Let (S,\Sigma,\mu) be a \sigma-finite measure space. Suppose that g \colon S \to [0,\infty] is a \Sigma-measurable function such that \int f \cdot g \; d \mu \leq \int f \; d \mu for every integrable function f \colon S \to [0,\infty). Prove that g \leq 1 \mu-almost everywhere on S.
    2. Let (S,\Sigma,\mu) be a \sigma-finite measure space. Let f \colon S \to (0,\infty) be a \Sigma-measurable function with \int f \; d \mu = 1 and let 0 < r < 1. Prove that for every E \in \Sigma with 0 < \mu(E) < \infty, \int f^r d \mu \leq \int \mu(E)^{1-r}.
    3. Let (S,\Sigma,\mu) be a \sigma-finite measure space. Let g \colon S \to \mathbb{R} be integrable. Prove that there exists a bounded \Sigma-measurable function f \colon S \to \mathbb{R} such that \lVert f \rVert_\infty = 1 and \int f \cdot g \; d \mu = \lVert g \rVert_1.
    4. If f \in L^{\infty}, prove that |f| \leq \lVert f \rVert_\infty \mu-almost everywhere. Moreover, prove that if \alpha < \lVert f \rVert_\infty, then there exists an E \in \Sigma with \mu(E) > 0 and |f(s)| \geq \alpha for all s \in E.
    5. Prove that (L^\infty (S,\Sigma,\mu), \lVert \cdot \rVert_\infty) is a normed linear space.
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  2. #2
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    Oct 2008
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    Homework Help??

    I did not realize that Miami University graduate students in the department of mathematics turned to the internet for homework help 2 days before it is due.
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