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

• Jan 16th 2012, 05:06 PM
jj323
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?
• Jan 17th 2012, 01:00 AM
girdav
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?
• Feb 7th 2014, 10:22 AM
jj323
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