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

Hi,

Under what conditions does the following equality hold

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

where $\displaystyle \mu$ denotes some measure?

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 $\displaystyle \Omega\to \{x\}$ mean?

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