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

[peb78]

Hi. Under what conditions does the following equality hold?

$f(x)=\lim\limits_{\Omega\rightarrow\{x\}} \frac{1}{\mu(\Omega)}\int_\Omega f d\mu$

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

$\bigcap\limits_{i=1}^{\infty} \Omega_i=\{x\}$.

Then, define the consider the sequence $\{y_i\}_{i=1}^\infty$ where

$y_i=\frac{1}{\mu(\Omega_i)}\int_{\Omega_i} f d\mu$

Under what conditions does $\lim\limits_{i\rightarrow\infty} y_i=f(x)$?

 

[micromass]

Check out http://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem

 

[peb78]

That's exactly what I'm looking for. Thanks!