Hello, I am stuck with the following problem.

Let $\displaystyle \Omega$ be the unit square $\displaystyle \Omega=\{(x,y):0<x,y \leq 1\}$, let $\displaystyle \mathcal{F}$ be the class of sets of the form $\displaystyle \{(x,y): x \in A, 0 < y \leq 1\}$, where $\displaystyle A \in \mathcal{B}$.

($\displaystyle \mathcal{B}=\sigma(\mathcal{B}_0)$ and $\displaystyle \mathcal{B}_0$ is the class of finite disjoint unions of subintervals of $\displaystyle (0,1]$ augmented by the empty set). Let $\displaystyle P=\lambda(A)$ at this set(for a subinterval $\displaystyle I=(a,b]$ of $\displaystyle (0,1]$, $\displaystyle \lambda(I)=|I|=b-a$). Show that $\displaystyle (\Omega,\mathcal{F},P)$ is a probability measure space. Show that for $\displaystyle A=\{(x,y):0<x \leq 1, y=1/2 \}$ that $\displaystyle P_*(A) = 0$ and$\displaystyle P^*(A) = 1$ ($\displaystyle P^*$ is the outer measure of $\displaystyle P$ and $\displaystyle P_*$ is the inner measure of $\displaystyle P$)

Thanks in advance,