# Finding a counterexample for inequality of measures

Let $\mu,\nu$ be measures on a set $\mathcal E \subseteq \mathcal P(X)$ and $\sigma(\mathcal E)$ its generated sigma algebra. I would now like to find a counterexample to the following proposition (I know it is wrong) :

"For $\mu_o := \mu|_{\mathcal E}$ and $\nu_o := \nu|_{\mathcal E}$ , if $\mathcal E$ is closed under intersections and $\mu_0(A) \le \nu_0(A)$ for all $A\in \mathcal E$ then $\mu \leq \nu$."

I know that this proposition is true, if $\mathcal E$ denoted a semiring. However, I am struggling to find a counterexample for this one. I was thinking of $\mathcal E= \{\{2\}, \{2,3\},\{1,2\}\}$ and $X = \{1,2,3\}$. Here I am getting stuck. I need to find two measures but I cannot seem to find ones that make me disprove the statement above. Any help is greatly appreciated!

How about $X = \{1,2\}$ and $\mathcal{E} = \left\{ \{1\}, \{1,2\} \right\}$.
You could take $\mu(X) = \nu(X) = 1$, $\mu(\{1\}) = 0$, and $\nu(\{1\}) = 1$.