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

1. Jan 18, 2012

### 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 $\lim\limits_{i\rightarrow\infty} y_i=f(x)$?

2. Jan 18, 2012

### micromass

Staff Emeritus
3. Jan 18, 2012

### peb78

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