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?
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?
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?
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