## Probability Measure Space Problem

Hello, I am stuck with the following problem.

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