Example of $\mu_1,\mu_2$ where $\mu_{1}(F)=\mu_{2}(F)\, \forall F \in \mathcal{F}$ but $\mu_1\ne \mu_2$

Question

Suppose $$\mu_{1}:\sigma(\mathcal{F})\rightarrow [0,1]$$ $$\mu_{2}:\sigma(\mathcal{F})\rightarrow [0,1] $$ are probability measures. And $\forall F \in \mathcal{F}$, $\mu_{1}(F)=\mu_{2}(F)$. But $\mu_{1}\neq \mu_{2} $. I need to give an example of $\mu_{1}, \mu_{2}$.

I've spend much time and have no idea. Please give me any hints...

Answer 1

Take sets $A = \{1,2\},B=\{1,3\}$ be a subset of $\{1,2,3,4\}$. Let $\mathcal F = \{A,B\}$. Then define $\mu_1,\mu_2$ such that:

$$\mu_1(\{2\}) = \mu_1(\{3\}) = \frac{1}{2}$$

and

$$ \mu_2(\{1\}) = \mu_2(\{4\}) = \frac{1}{2} $$

Then $\mu_1(A) = \mu_1(B) = \frac{1}{2} = \mu_2(A) = \mu_2(B)$

But $\mu_1(\{1\}) = 0 \not = \frac{1}{2} = \mu_2(\{1\}) $