$\text{Let } X= [0,1] \times [0,1] \text{ and define:}$
$S=\bigl\{R \text{ | } R=[0,1]\times[a,b] \text{ or } R=[a,b]\times[0,1]\bigl\}$
$\text{We define the function on } S: m(R)=\text{Area}(R)$
$\text {Show that the measure } m \text { has more than one expansion.}$
I am not sure if I understand the question but I will try anyway to given an answer. Consider the map $\tau :[0,1] \to \mathbb R^{2}$ defined by $\tau (x)=(x,x)$ and let $\mu =m\circ \tau ^{-1}$, i.e. $\mu (E) =m\{x\in [0,1]:(x,x) \in E\}$. Let $\nu $ be Lebesgue measure in $\mathbb R^{2}$. Then both $\mu$ and $\nu $ assign the same value, ($b-a$), to the rectangles under consideration so there are multiple extensions.