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

1. ## 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?

2. ## 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?

3. ## 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