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Math Help - Measuring the reals

  1. #1
    MHF Contributor Swlabr's Avatar
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    Measuring the reals

    Let (\mathbb{R}, \Sigma, \mu) be a measure space, \mu some measure. There exists at least one Lebesgue integrable function. Then does it hold that,

    \mu(\mathbb{R}) < \infty \Rightarrow \mu(\mathbb{R}) = 0?
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  2. #2
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    What exactly are you asking? As for the last part you can just take \mu to be the zero measure (not interesting but a measure none the less).
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by putnam120 View Post
    What exactly are you asking? As for the last part you can just take \mu to be the zero measure (not interesting but a measure none the less).
    It's okay - I found an example (although I can't remember what it is...).

    Basically, my question was: does there exist a (positive) measure such that \mu(\mathbb{R}) < \infty but which is not the zero measure.
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  4. #4
    Moo
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    Hello,

    Well just make \mu a Dirac measure
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  5. #5
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    Let \nu be any measure defined on \mathbb{R} such that there exists A\in \Sigma with 0<\nu(A)<\infty. Then the measure \mu(E)=\nu(E\cap A) will be finite.
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