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Math Help - Measure theoretic integral of a function over a domain shrinking to a singleton set

  1. #1
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    Measure theoretic integral of a function over a domain shrinking to a singleton set

    Hi,
    Under what conditions does the following equality hold

    f(x)=\lim_{\Omega\rightarrow \{x\}} \frac{1}{\mu(\Omega)}\int_{\Omega} f d\mu

    where \mu denotes some measure?
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  2. #2
    Super Member girdav's Avatar
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    Re: Measure theoretic integral of a function over a domain shrinking to a singleton s

    Maybe you should add some precisions: what is fixed, and on what are the conditions you are looking for: on the function, the measured space? And what does \Omega\to \{x\} mean?
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  3. #3
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    Re: Measure theoretic integral of a function over a domain shrinking to a singleton s

    This is a little bit old, but in case someone stumbles upon the question, this is the Lebesgue differentiation theorem:

    https://en.wikipedia.org/wiki/Lebesg...iation_theorem
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