peb78
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Hi. Under what conditions does the following equality hold?
[itex]f(x)=\lim\limits_{\Omega\rightarrow\{x\}} \frac{1}{\mu(\Omega)}\int_\Omega f d\mu[/itex]
where [itex]\mu[/itex] is some measure. Being a little more careful, let [itex]\Omega_i[/itex] be a sequence of sets such that [itex]\Omega_{i+1}\subseteq\Omega_i[/itex] and
[itex]\bigcap\limits_{i=1}^{\infty} \Omega_i=\{x\}[/itex].
Then, define the consider the sequence [itex]\{y_i\}_{i=1}^\infty[/itex] where
[itex]y_i=\frac{1}{\mu(\Omega_i)}\int_{\Omega_i} f d\mu[/itex]
Under what conditions does [itex]\lim\limits_{i\rightarrow\infty} y_i=f(x)[/itex]?
[itex]f(x)=\lim\limits_{\Omega\rightarrow\{x\}} \frac{1}{\mu(\Omega)}\int_\Omega f d\mu[/itex]
where [itex]\mu[/itex] is some measure. Being a little more careful, let [itex]\Omega_i[/itex] be a sequence of sets such that [itex]\Omega_{i+1}\subseteq\Omega_i[/itex] and
[itex]\bigcap\limits_{i=1}^{\infty} \Omega_i=\{x\}[/itex].
Then, define the consider the sequence [itex]\{y_i\}_{i=1}^\infty[/itex] where
[itex]y_i=\frac{1}{\mu(\Omega_i)}\int_{\Omega_i} f d\mu[/itex]
Under what conditions does [itex]\lim\limits_{i\rightarrow\infty} y_i=f(x)[/itex]?