# Sigma-algebra

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• Sep 26th 2011, 11:36 AM
kierkegaard
Sigma-algebra
Let $A$ the collection of finite unions of sets of the form
$(a,b]$ $\cap$ $\mathbb{Q}$, when $-\infty$ $\leq$a<b $\leq$ $+\infty$."

a. Show that $A$ is an algebra on $\mathbb{Q}$.

b. Show that the $\sigma$-algebra generated by $A$ is
P( $\mathbb{Q}$) (the power set of $\mathbb{Q}$).

c. Define $\nu$ on $A$ by $\nu$( $\phi$)=0 and $\nu$(C)= $+\infty$ for C $\neq$ $phi$. Show that $\nu$ is a premeasure on $A$ and that there is more than one measure on P( $\mathbb{Q}$) whose restriction to $A$ is $\nu$.

Thanks
• Sep 26th 2011, 12:38 PM
FernandoRevilla
Re: Sigma-algebra
It would be convenient to show some work. What difficulties have you had?. For example $\mathbb{Q}=(-\infty,+\infty)\cap \mathbb{Q}$ so, $\mathbb{Q}\in A$ .
• Sep 26th 2011, 12:45 PM
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, 01:03 PM
kierkegaard
Re: Sigma-algebra
I just solved the problem. It was quite simple :/. Sorry for your time.