peb78
Jan18-12, 09:41 AM
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)?
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)?