Exer Let $\mathbb{P}$ be the uniform (Lebesgue) measure on $\Omega=[0,1]$. Define
$$
Z(\omega)=\left\{\begin{array}{l}
0 \text { if } 0 \leq \omega<\frac{1}{2}, \\
2 \text { if } \frac{1}{2} \leq \omega \leq 1 .
\end{array}\right.
$$
For $A \in \mathcal{B}[0,1]$, define
$$
\widetilde{\mathbb{P}}(A)=\int_A Z(\omega) d \mathbb{P}(\omega) .
$$
(i) Show that $\widetilde{\mathbb{P}}$ is a probability measure.
(ii) Show that if $\mathbb{P}(A)=0$, then $\widetilde{\mathbb{P}}(A)=0$. We say that $\widetilde{\mathbb{P}}$ is absolutely continuous with respect to $\mathbb{P}$.
(iii) Show that there is a set $A$ for which $\widetilde{\mathbb{P}}(A)=0$ but $\mathbb{P}(A)>0$. In other words, $\widetilde{\mathbb{P}}$ and $\mathbb{P}$ are not equivalent.