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