Sigma-algebra

**kierkegaard** · September 26th 2011, 10:36 AM

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

**FernandoRevilla** · September 26th 2011, 11:38 AM

It would be convenient to show some work. What difficulties have you had?. For example $\mathbb{Q}=(-\infty,+\infty)\cap \mathbb{Q}$ so, $\mathbb{Q}\in A$ .

**kierkegaard** · September 26th 2011, 11:45 AM

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.

**kierkegaard** · September 26th 2011, 12:03 PM

I just solved the problem. It was quite simple :/. Sorry for your time.

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