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Value of a measure theoretic integral over a domain shrinking to a single set

  1. Jan 18, 2012 #1
    Hi. Under what conditions does the following equality hold?

    [itex]f(x)=\lim\limits_{\Omega\rightarrow\{x\}} \frac{1}{\mu(\Omega)}\int_\Omega f d\mu[/itex]

    where [itex]\mu[/itex] is some measure. Being a little more careful, let [itex]\Omega_i[/itex] be a sequence of sets such that [itex]\Omega_{i+1}\subseteq\Omega_i[/itex] and

    [itex]\bigcap\limits_{i=1}^{\infty} \Omega_i=\{x\}[/itex].

    Then, define the consider the sequence [itex]\{y_i\}_{i=1}^\infty[/itex] where

    [itex]y_i=\frac{1}{\mu(\Omega_i)}\int_{\Omega_i} f d\mu[/itex]

    Under what conditions does [itex]\lim\limits_{i\rightarrow\infty} y_i=f(x)[/itex]?
     
  2. jcsd
  3. Jan 18, 2012 #2

    micromass

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  4. Jan 18, 2012 #3
    That's exactly what I'm looking for. Thanks!
     
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