# Sigma-algebra

• Sep 26th 2011, 10:36 AM
kierkegaard
Sigma-algebra
Let $\displaystyle A$ the collection of finite unions of sets of the form
$\displaystyle (a,b]$$\displaystyle \cap$$\displaystyle \mathbb{Q}$, when $\displaystyle -\infty$ $\displaystyle \leq$a<b$\displaystyle \leq$$\displaystyle +\infty." a. Show that \displaystyle A is an algebra on \displaystyle \mathbb{Q}. b. Show that the \displaystyle \sigma-algebra generated by \displaystyle A is P(\displaystyle \mathbb{Q}) (the power set of \displaystyle \mathbb{Q}). c. Define \displaystyle \nu on \displaystyle A by \displaystyle \nu(\displaystyle \phi)=0 and \displaystyle \nu(C)=\displaystyle +\infty for C \displaystyle \neq$$\displaystyle phi$. Show that $\displaystyle \nu$ is a premeasure on $\displaystyle A$ and that there is more than one measure on P($\displaystyle \mathbb{Q}$) whose restriction to $\displaystyle A$ is $\displaystyle \nu$.

Thanks
• Sep 26th 2011, 11:38 AM
FernandoRevilla
Re: Sigma-algebra
It would be convenient to show some work. What difficulties have you had?. For example $\displaystyle \mathbb{Q}=(-\infty,+\infty)\cap \mathbb{Q}$ so, $\displaystyle \mathbb{Q}\in A$ .
• Sep 26th 2011, 11:45 AM
kierkegaard
Re: Sigma-algebra
I have problems to prove (b); one of the implications is obvious because the sigma-algebra generated by A is contained in P(Q), but I don't know how to prove the other direction. Thanks.
• Sep 26th 2011, 12:03 PM
kierkegaard
Re: Sigma-algebra
I just solved the problem. It was quite simple :/. Sorry for your time.