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Math Help - A characterization of an almost everywhere constant variable

  1. #1
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    A characterization of an almost everywhere constant variable

    I've been thinking for several hours now, and can't see it.

    Let ([0,1],\mathcal{M},\mathbb{P}) be a probability space. Let f:[0,1]\rightarrow\mathbb{R} be a \mathbb{P}-integrable variable (in the Lebesgue sense). Then the following conditions are equivalent:

    - f is \mathbb{P}-almost everywhere constant
    - e^{\int_{[0,1]}f(x)\mathrm{d}\mathbb{P}}=\int_{[0,1]}e^{f(x)}\mathrm{d}\mathbb{P}

    Obviously I must be missing some inequality, probably well known. Any help will be appreciated.
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  2. #2
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    This is an attainment of Jensen's Inequality.
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  3. #3
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    Thank you very much.
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